Solution:
First, let's consider a section $\mathcal{P}$ of the solids, along the axis.
By some 3D-Geomerty thinking, we can simply know that the axis crosses the sphere center. So, that is saying, the $\mathcal{P}$ we took crosses one of the equator of the sphere.
Here I drew two graphs, the first one is the case when $T$ is internally tangent to $S$,

and the second one is when $T$ is externally tangent to $S$.

For both graphs, point $O$ is the center of sphere $S$, and points $A$ and $B$ are the intersections of the sphere and the axis. Point $E$ (ignoring the subscripts) is one of the circle centers of the intersection of torus $T$ with section $\mathcal{P}$. Point $G$ (again, ignoring the subscripts) is one of the tangents between the torus $T$ and sphere $S$ on section $\mathcal{P}$. $EF\bot CD$, $HG\bot CD$.
And then, we can start our calculation.
In both cases, we know $\Delta OEF\sim \Delta OGH\Longrightarrow \frac{EF}{OE} =\frac{GH}{OG}$.
Hence, in the case of internal tangent, $\frac{E_iF_i}{OE_i} =\frac{G_iH_i}{OG_i}\Longrightarrow \frac{6}{11-3} =\frac{r_i}{11}\Longrightarrow r_i=\frac{33}{4}$.
In the case of external tangent, $\frac{E_oF_o}{OE_o} =\frac{G_oH_o}{OG_o}\Longrightarrow \frac{6}{11+3} =\frac{r_o}{11}\Longrightarrow r_o=\frac{33}{7}$.
Thereby, $r_i-r_o=\frac{33}{4}-\frac{33}{7}=\frac{99}{28}$. And there goes the answer, $99+28=\boxed{\mathbf{127} }$
~Prof_Joker

Answer: 127

<｜begin▁of▁sentence｜><｜User｜>Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the outside of $S$, it is externally tangent to $S$ along a circle with radius $r_i$, and when $T$ rests on the outside of $S$, it is externally tangent to $S$ along a circle with radius $r_o$. The difference $r_i-r_o$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.<｜Assistant｜><think>

Okay, so I need to solve this problem about a torus and a sphere. Let me first try to visualize it. The torus T is created by rotating a circle with radius 3 around an axis in its plane, which is 6 units away from the center of the circle. That makes sense—it's like a donut shape where the distance from the center of the generating circle to the axis is 6, and the radius of the generating circle is 3.

Then there's a sphere S with radius 11. The problem says when the torus rests on the outside of the sphere, it's externally tangent along a circle with radius r_i, and when it rests on the inside, it's externally tangent along a circle with radius r_o. Wait, actually, hold on. The wording says "when T rests on the outside of S, it is externally tangent to S along a circle with radius r_i", and similarly for the outside. Hmm, maybe I need to parse that again.

Wait, actually, maybe it's a translation error? Let me check again: "When T rests on the outside of S, it is externally tangent to S along a circle with radius r_i, and when T rests on the outside of S, it is externally tangent to S along a circle with radius r_o." Wait, that can't be right. Maybe the second one is when it rests on the inside? Let me check the original problem again.

Wait, the user wrote: "When T rests on the outside of S, it is externally tangent to S along a circle with radius r_i, and when T rests on the outside of S, it is externally tangent to S along a circle with radius r_o." Hmm, that seems like a typo. Maybe the second part is when it rests on the inside? Because otherwise, both are resting on the outside. Let me check the original problem again as presented by the user.

Wait, the user wrote: "When T rests on the outside of S, it is externally tangent to S along a circle with radius r_i, and when T rests on the outside of S, it is externally tangent to S along a circle with radius r_o." Hmm, that must be a mistake. Probably, one is resting on the inside and the other on the outside. Maybe the second one is resting on the inside? Let me see the exact problem statement again.

Wait, the user's exact words: "When T rests on the outside of S, it is externally tangent to S along a circle with radius r_i, and when T rests on the outside of S, it is externally tangent to S along a circle with radius r_o." Hmm, that's strange. Maybe the user made a typo. Alternatively, perhaps "rests on the outside" and "rests on the inside" but the problem as stated has a typo. Alternatively, maybe "externally tangent" and "internally tangent"? Wait, but the problem says "externally tangent" both times. Hmm, perhaps the problem is as stated, but maybe I need to think differently.

Alternatively, maybe when the torus is resting on the outside of the sphere, the tangent circle has radius r_o, and when it's resting on the inside, it's r_i? Wait, but the problem says "externally tangent" in both cases. Hmm, perhaps the problem is correct, but I need to figure out how a torus can rest on the outside of a sphere and be externally tangent. Maybe "externally tangent" here means that the torus is outside the sphere, touching it along a circle, and similarly when it's resting on the inside, it's still externally tangent? Hmm, maybe the terminology is different. Alternatively, perhaps "externally tangent" is when the torus is outside the sphere, and "internally tangent" would be when it's inside, but the problem says "externally tangent" in both cases. Hmm, maybe I need to proceed with the problem as stated, assuming that there are two different configurations where the torus is externally tangent to the sphere along a circle, one when it's resting on the outside and one when it's resting on the inside? Maybe "resting on the outside" and "resting on the inside" are two different positions where the torus is externally tangent, but in different orientations.

Alternatively, perhaps the problem is that when the torus is placed outside the sphere, the circle of tangency has radius r_o, and when it's placed inside, the circle has radius r_i, but the problem says "externally tangent" in both cases. Hmm, maybe "externally tangent" just means that they are tangent from the outside, so when the torus is outside the sphere, the tangent circle is on the outer side, and when it's inside, the tangent is on the inner side? Not sure. Maybe I need to proceed with the problem as stated, even if there's some confusion in the wording. Let me try to think step by step.

First, let's recall some properties of a torus. A torus can be described by two radii: the distance from the center of the generating circle to the axis of rotation (which is 6 in this case), and the radius of the generating circle (which is 3 here). So, the torus has major radius R = 6 and minor radius r = 3.

Now, the sphere has radius 11. We need to consider two configurations where the torus is externally tangent to the sphere along a circle. The problem refers to "resting on the outside" and "resting on the outside" again, but that must be a mistake. Probably, one is resting on the outside and the other on the inside. Let me assume that the first case is when the torus is outside the sphere, tangent along a circle with radius r_o, and the second case is when the torus is inside the sphere, tangent along a circle with radius r_i. Then the difference r_i - r_o would be required. Alternatively, maybe the other way around. Since the problem says "externally tangent" in both cases, perhaps "externally tangent" just means that the torus is outside the sphere, so when it's resting on the outside, it's externally tangent (so r_o is the radius of the tangent circle), and when it's resting on the inside, maybe it's still externally tangent but in a different orientation? Hmm, perhaps I need to think in terms of the positions.

Alternatively, maybe "externally tangent" is a standard term here. Let me recall that for two surfaces, external tangent means they lie on opposite sides of the tangent plane, while internal tangent means they lie on the same side. But in 3D, tangency along a circle would mean that they share a common tangent plane along that circle. Hmm, perhaps the key is to find the radii of the circles where the torus and sphere are tangent in these two different configurations.

Alternatively, perhaps the problem is that when the torus is placed so that it's enclosing the sphere, the tangent circle has radius r_i, and when it's placed so that the sphere is enclosing the torus, the tangent circle has radius r_o. But the sphere has radius 11, and the torus has a major radius of 6 plus a minor radius of 3, so the maximum distance from the center of the torus to any point on the torus is 6 + 3 = 9, which is less than 11. Therefore, the sphere is larger. So, the torus can rest on the outside of the sphere, meaning the sphere is inside the torus? Wait, but the sphere is bigger. Wait, the sphere has radius 11, so if the torus is placed around the sphere, since the torus's maximum radius is 9, which is smaller than 11, the sphere would extend beyond the torus. Hmm, maybe I need to think differently.

Alternatively, perhaps the torus is placed such that it is externally tangent to the sphere, meaning they touch at a circle, and the torus is outside the sphere. Since the sphere is larger, the torus can be placed outside the sphere, touching it along a circle. Alternatively, the torus can be placed inside the sphere, touching it along another circle. So, in both cases, they are externally tangent? Maybe "externally tangent" just means that they are tangent but not intersecting otherwise, so in both cases, the torus is either outside or inside the sphere, but tangent along a circle.

Therefore, perhaps the two cases are:

1. The torus is outside the sphere, touching it along a circle (so the sphere is inside the torus's "hole"), but since the sphere is larger, maybe this isn't possible? Wait, the sphere has radius 11, and the torus's central hole has a radius of 6 - 3 = 3. So the hole is only radius 3, so the sphere can't fit inside the hole. Therefore, perhaps when the torus is resting on the outside of the sphere, meaning the sphere is inside the torus's main body? Hmm, this is getting confusing. Maybe I need to approach this mathematically.

Let me recall that the equation of a torus can be given in cylindrical coordinates (r, θ, z) as:

√( (R - √(x² + y²))² + z² ) = r

Wait, actually, the standard equation for a torus centered at the origin, with the z-axis as the axis of rotation, is:

(√(x² + y²) - R)^2 + z² = r²,

where R is the major radius (distance from the center of the generating circle to the axis) and r is the minor radius (radius of the generating circle). So in this case, R = 6 and r = 3. So the equation is (√(x² + y²) - 6)^2 + z² = 9.

The sphere is centered somewhere. Wait, but the problem doesn't specify the position of the sphere. Wait, probably, when the torus is resting on the sphere, they are tangent, so their centers must lie along the same line. Since the torus is symmetric around its axis, which we can assume is aligned with the sphere's center. Let me assume that both the torus and the sphere are centered along the same axis, say the z-axis. Wait, but the torus's axis is the z-axis, so the sphere's center must be along the z-axis as well.

Wait, but when the torus is resting on the sphere, the point of tangency would be along a circle. Let me think of the cross-sectional view. If we take a cross-section through the z-axis, the torus would look like a circle of radius 6 (the center of the generating circle) with a circle of radius 3 around a point 6 units from the origin. The sphere, in cross-section, would be a circle of radius 11.

So in cross-section, the problem reduces to two circles: one with radius 11 (the sphere) and the other is the cross-section of the torus, which is a circle of radius 3 centered at (6, 0). Wait, actually, in cross-section, the torus would appear as two circles: one at (6,0) and one at (-6,0), each with radius 3. But since we are considering the torus as a surface of revolution, perhaps in cross-section, the relevant circle is the one on the same side as the sphere.

Wait, perhaps it's better to model the problem in 3D. Let me consider the centers. Let me suppose that the sphere is centered at the origin (0,0,0). The torus is arranged such that its axis of rotation is along the z-axis, so its center (the center of the generating circle) is at (6,0,0) when rotated around the z-axis. Wait, actually, the torus is generated by rotating a circle of radius 3 around the z-axis, but the circle itself is in a plane perpendicular to the z-axis. Wait, actually, the standard torus has its generating circle in the plane perpendicular to the axis of rotation. So if the axis of rotation is the z-axis, then the generating circle lies in the plane z = 0, centered at (R, 0, 0) where R is the major radius. So in this case, the generating circle is centered at (6, 0, 0) with radius 3. So the torus is the surface of all points obtained by rotating this circle around the z-axis.

Therefore, the torus's equation is (√(x² + y²) - 6)^2 + z² = 9.

Now, the sphere is of radius 11. Let's assume the sphere is centered at the origin (0,0,0). Then, to find the points where the torus and the sphere are tangent, we can set up equations for their intersection and impose the tangency condition.

But the problem mentions two different configurations: when the torus rests on the outside of the sphere and when it rests on the inside. Wait, but if the sphere is fixed at the origin, how can the torus rest on the outside or inside? Maybe the torus is moved so that it is tangent to the sphere in two different ways.

Alternatively, perhaps the torus is fixed, and the sphere is placed such that it is tangent to the torus either from the outside or the inside. Hmm, but the problem says "when T rests on the outside of S", so the torus is resting on the sphere. So perhaps the sphere is fixed, and the torus is placed on top of it, so that they are tangent along a circle. Depending on whether the torus is placed on the outside (enveloping the sphere?) or on the inside (sitting inside the sphere?), but given the sphere is larger, maybe the torus can be placed inside the sphere, touching it along a circle, or placed outside, but since the sphere is bigger, perhaps the torus can only be placed inside? Hmm, this is confusing.

Alternatively, perhaps the sphere is not fixed at the origin. Maybe both the torus and the sphere can be positioned such that they are tangent along a circle, and in one case the torus is outside the sphere, and in the other case, it's inside. The key is that in both cases, they are externally tangent, meaning they touch at a single circle without intersecting elsewhere.

Alternatively, perhaps "externally tangent" here means that the torus and sphere are tangent along a circle, and the torus is outside the sphere in one case and inside in the other, but the term "externally tangent" is used in both cases. Maybe the problem is using "externally tangent" to mean that they are tangent but not intersecting otherwise, so in both configurations (torus outside and inside the sphere), they are externally tangent along a circle.

To proceed, perhaps I need to find the radii r_i and r_o of the circles of tangency in the two different configurations where the torus is tangent to the sphere, one where the torus is closer to the sphere's center (inside) and one where it's farther (outside). Wait, but how can the torus be outside the sphere? Since the sphere has radius 11, and the torus's maximum distance from its own center is 6 + 3 = 9, so if the torus is centered at the same point as the sphere, then the torus would be entirely inside the sphere. Therefore, to have the torus resting on the outside of the sphere, perhaps the torus is shifted away from the sphere's center so that part of it extends outside? Hmm, but the problem might assume that both the torus and sphere are centered at the same point. Wait, but if they are centered at the same point, then the torus is entirely inside the sphere, since the sphere has a larger radius.

Alternatively, maybe the torus is not centered at the sphere's center. Let me think. Let me suppose that the sphere is centered at the origin, and the torus is shifted along the z-axis so that it is tangent to the sphere. Then, depending on the direction of the shift, the torus can be either above or below, but since it's a surface of revolution, it's symmetric. Alternatively, shifting along the x-axis. Wait, perhaps the centers are aligned along the same line. Let me try to formalize this.

Let me denote the center of the sphere as point O, and the center of the generating circle of the torus as point C. Since the torus is generated by rotating a circle around an axis, the axis is the line perpendicular to the plane of the generating circle, passing through the center of rotation. In our case, the generating circle is in the plane perpendicular to the z-axis, centered at (6, 0, 0). Wait, but if the sphere is at the origin, then the distance between the sphere's center and the generating circle's center is 6 units along the x-axis.

Wait, perhaps the key is to consider the distance between the centers of the sphere and the torus. Let me think of the torus as a surface, so its "center" is the center of the generating circle, which is at (6,0,0). The sphere is at the origin. The distance between these centers is 6 units.

To find the points where the torus and sphere intersect tangentially, we can consider the points on both surfaces where their normals are colinear, meaning the gradient vectors of their equations point in the same direction.

Alternatively, perhaps it's easier to consider the cross-sectional view. Let's take a cross-section in the plane of the generating circle, say the x-z plane. In this cross-section, the torus appears as a circle of radius 3 centered at (6, 0). The sphere appears as a circle of radius 11 centered at (0,0).

Wait, in cross-section, the torus would actually be represented by two circles: one at (6,0) with radius 3 and another at (-6,0) with radius 3, but since we are considering the generating circle in the x-z plane, perhaps only the right one is relevant here.

So in this 2D cross-section, the problem reduces to finding the points where the circle of radius 11 (sphere) and the circle of radius 3 centered at (6,0) (generating circle of the torus) are tangent. Wait, but in 3D, the tangency is along a circle, which in cross-section would be a point. Hmm, maybe this approach is not directly applicable.

Alternatively, perhaps the circle of tangency lies in a plane. Let me consider that the torus and sphere are tangent along a circle lying in some plane. The centers of the sphere and the torus must lie along the line perpendicular to this plane. Since the torus is symmetric around its axis (the z-axis in our setup), the plane of tangency must be perpendicular to the z-axis or parallel? Wait, perhaps the plane of tangency is perpendicular to the line connecting the centers of the sphere and the torus.

Let me denote the center of the sphere as O (0,0,0) and the center of the generating circle of the torus as C (6,0,0). The line connecting O and C is along the x-axis. Therefore, the plane of tangency must be perpendicular to the x-axis, so it's a plane parallel to the y-z plane. The circle of tangency lies in such a plane. Let me suppose that the plane is x = k for some k between 0 and 6 (if the torus is between the sphere and the origin) or beyond 6 (if the torus is outside).

Wait, but the sphere is radius 11, so if the torus is centered at (6,0,0), the distance from the sphere's center to the torus's center is 6. The sphere's radius is 11, so the torus is well within the sphere. Therefore, if we want the torus to be externally tangent to the sphere, perhaps we need to move the torus away from the sphere's center? Wait, but the problem says "when T rests on the outside of S", so maybe the torus is placed such that it's outside the sphere, but how?

Alternatively, perhaps the problem is considering the torus and sphere with their centers separated by some distance, and in one configuration, the torus is outside the sphere (so the distance between centers plus the torus's size equals the sphere's radius?), but this is getting confusing. Maybe I need to set up equations.

Let me consider the general case where the sphere is centered at the origin, and the torus's generating circle is centered at (d, 0, 0), so the major radius R = d, and minor radius r = 3. Wait, but in the problem, R is given as 6, so d = 6. The sphere has radius 11.

We need to find the distance between the centers (which is d = 6) such that the torus and sphere are tangent along a circle. Wait, but if the centers are fixed at distance 6 apart, then perhaps the tangency condition can be satisfied in two different ways?

Alternatively, maybe the torus can be placed in two different positions relative to the sphere where they are tangent along a circle. For example, the torus could be placed so that the sphere is inside the "hole" of the torus, or outside. But since the sphere is larger, it can't fit inside the hole (the hole has radius 6 - 3 = 3, which is much smaller than the sphere's radius 11). Therefore, the sphere must be outside the torus. Wait, but the torus's maximum distance from its center is 6 + 3 = 9, so if the sphere is centered 6 units away, the closest point on the torus to the sphere's center is 6 - 9 = -3? Wait, no. Let me think again.

Wait, the distance from the sphere's center (origin) to the torus's generating circle center is 6. The closest point on the torus to the origin would be along the line connecting the two centers, so towards the origin. The generating circle is at (6,0,0) with radius 3, so the closest point on the generating circle is at (6 - 3, 0, 0) = (3,0,0). Therefore, the closest point on the torus to the origin is 3 units away from the generating circle's center towards the origin, so at distance 6 - 3 = 3 from the origin. Wait, but the sphere has radius 11, so the sphere would encompass the entire torus. Therefore, if the torus is fixed with its generating circle center at 6 units from the sphere's center, then the torus is entirely inside the sphere. Therefore, to have them tangent, perhaps we need to move the torus away from the sphere's center?

Wait, maybe the problem is not assuming the centers are fixed. Maybe the torus can be moved so that it is tangent to the sphere. So the distance between the centers is variable, and we need to find the positions where the torus is tangent to the sphere along a circle, in two different configurations (inside and outside).

Let me rephrase the problem: given a torus with major radius R = 6 and minor radius r = 3, and a sphere of radius 11, find the radii r_i and r_o of the circles of tangency when the torus is placed such that it is externally tangent to the sphere in two different configurations (probably one where the torus is closer to the sphere's center and one where it's farther away), then compute r_i - r_o.

To model this, let's consider the centers of the sphere (O) and the generating circle of the torus (C). Let the distance between O and C be D. The torus is generated by rotating a circle of radius 3 around the axis perpendicular to OC at point C. Wait, actually, the axis of the torus is the line perpendicular to the plane of the generating circle. Since the generating circle is in the plane perpendicular to the axis. If the generating circle is centered at C and the axis is perpendicular to OC, then the axis is along the line perpendicular to OC at point C. Hmm, perhaps it's better to set up coordinates so that O is at (0,0,0), and C is at (D, 0, 0). Then the axis of the torus is the line perpendicular to the x-axis at point C, so along the y-z plane at x = D.

The equation of the torus in this coordinate system would be: for any point (x, y, z), the distance from the axis (x = D, y=0, z=0) must satisfy the torus equation. The distance from a point (x,y,z) to the axis is sqrt( (x - D)^2 + y^2 + z^2 ). Wait, no, the axis is the line x = D, y=0, z=0? Wait, actually, the axis is the line perpendicular to the generating circle's plane. Since the generating circle is in the plane perpendicular to the axis. If the generating circle is centered at (D,0,0) and lies in the plane perpendicular to the axis, which is along the x-axis, then the axis is the x-axis itself? Wait, maybe I need to clarify.

Wait, perhaps the axis of the torus is the x-axis. Then the generating circle is in a plane perpendicular to the x-axis, say at position (D, 0, 0), so the generating circle is in the y-z plane at x = D, with radius 3. Therefore, the torus is formed by rotating this circle around the x-axis. Wait, but then the major radius R would be the distance from the generating circle's center to the axis, which is zero, since it's on the axis. That can't be. Wait, confusion here.

Wait, let me correct. The major radius R is the distance from the center of the generating circle to the axis of rotation. So if the generating circle is in the y-z plane at (D, 0, 0), and the axis of rotation is the x-axis, then the distance from the generating circle's center to the axis is zero, which would make it a horn torus, but in the problem, R is 6. Therefore, perhaps the axis is not the x-axis. Let me set this up properly.

Let me define the coordinate system so that the axis of the torus is the z-axis. Then the generating circle is in the plane perpendicular to the z-axis, say the x-y plane, centered at (R, 0, 0) where R is the major radius (6). The generating circle has radius r = 3. Therefore, the torus equation is (√(x² + y²) - R)^2 + z² = r².

Now, the sphere is centered at the origin (0,0,0) with radius 11. The problem is to find the radii of the circles where the torus is externally tangent to the sphere in two different configurations.

To find the points of tangency, we can consider that the torus and sphere must touch at a circle, so their equations must have a common solution set forming a circle, and their gradients must be colinear at those points.

Alternatively, perhaps we can consider the distance between the centers (which is R = 6 along the x-axis) and the sphere's radius, and use some geometric relations.

Alternatively, let's consider a cross-sectional view in the x-y plane. In this plane, the torus appears as two circles: one at (6,0) with radius 3, and the other at (-6,0) with radius 3 (but since we are in the x-y plane, the torus's cross-section is these two circles). The sphere in cross-section is a circle of radius 11 centered at (0,0).

The points of tangency between the sphere and the torus in this cross-section would be points where the sphere's circle and the torus's generating circle are tangent. Since the cross-section is 2D, the tangency points would be single points here, but in 3D, they form a circle.

Let me consider the generating circle in the x-y plane at (6,0) with radius 3. The sphere's cross-section is a circle of radius 11. The distance between their centers is 6. The points where they are tangent can be found by solving for the points where the two circles touch.

The two circles are:

1. Sphere's cross-section: x² + y² = 11² = 121.

2. Generating circle of torus: (x - 6)^2 + y² = 3² = 9.

To find their points of intersection, subtract the two equations:

(x² + y²) - [(x - 6)^2 + y²] = 121 - 9 =>

x² - (x² -12x + 36) = 112 =>

12x - 36 = 112 =>

12x = 148 => x = 148 / 12 = 37/3 ≈ 12.333...

But wait, the generating circle is centered at (6,0) with radius 3, so the maximum x-coordinate it can reach is 6 + 3 = 9. But 37/3 ≈ 12.333 is way beyond that. That can't be. So this suggests that in the cross-section, the two circles don't intersect, which makes sense because the sphere's cross-section is much larger.

Hmm, so perhaps my approach is wrong. Since in cross-section, the generating circle is entirely inside the sphere's cross-section circle, so they don't intersect. Therefore, the tangency must occur in another plane.

Alternatively, the circle of tangency is not in the x-y plane but in some other plane. Let me consider a general plane where the tangency occurs. Let me suppose that the circle of tangency lies in a plane perpendicular to the line connecting the centers of the sphere and the torus's generating circle. Since the sphere is at (0,0,0) and the torus's generating circle center is at (6,0,0), the line connecting them is along the x-axis. Therefore, the plane of tangency is perpendicular to the x-axis, so it's of the form x = k.

In this plane, the sphere's intersection is a circle of radius sqrt(11² - k²) in the y-z plane. The torus's intersection with the plane x = k is given by the torus equation:

(√(k² + y² + z²) - 6)^2 + ... Wait, no. Wait, the torus equation is (√(x² + y²) - 6)^2 + z² = 9.

Wait, in the plane x = k, substituting x = k into the torus equation gives:

(√(k² + y²) - 6)^2 + z² = 9.

But in the plane x = k, the coordinates are (k, y, z), so the equation becomes:

(√(k² + y²) - 6)^2 + z² = 9.

Hmm, this is a bit complicated. Let me consider the intersection curve between the torus and the sphere. For the surfaces to be tangent along a circle, their intersection must be a single circle, and the gradients must be colinear along that circle.

Alternatively, perhaps we can parametrize the problem. Let me consider a point on both the torus and the sphere. Let me denote the sphere's equation as x² + y² + z² = 11² = 121.

The torus equation is (√(x² + y²) - 6)^2 + z² = 9.

Let me subtract the two equations to find a relation. Let me expand the torus equation:

(√(x² + y²) - 6)^2 + z² = (√(x² + y²))² - 12√(x² + y²) + 36 + z² = (x² + y²) - 12√(x² + y²) + 36 + z² = 9.

So, (x² + y² + z²) - 12√(x² + y²) + 36 = 9.

But the sphere equation gives x² + y² + z² = 121, so substituting that in:

121 - 12√(x² + y²) + 36 = 9 =>

157 - 12√(x² + y²) = 9 =>

-12√(x² + y²) = 9 - 157 =>

-12√(x² + y²) = -148 =>

√(x² + y²) = 148 / 12 = 37/3 ≈ 12.333...

But √(x² + y²) is the radial distance in the x-y plane, so this would mean that at the points of intersection, the x-y radius is 37/3. However, substituting back into the sphere's equation, the z-coordinate would be sqrt(121 - (37/3)^2 - y²), but this seems problematic because (37/3)^2 is (1369/9) ≈ 152.11, which is already larger than 121, so 121 - 1369/9 would be negative. Therefore, this is impossible.

Hmm, so this suggests that there is no real intersection between the torus and the sphere when their centers are 6 units apart? But that contradicts the problem statement which says they are tangent. Therefore, my assumption that the centers are fixed 6 units apart must be wrong.

Ah! Wait a second. The problem says "when T rests on the outside of S", so perhaps the distance between the centers is not fixed. The torus can be moved so that it is tangent to the sphere. The major radius of the torus is 6, but that is the distance from its generating circle's center to its axis. The position of the torus relative to the sphere can vary.

Therefore, perhaps I need to consider the distance between the sphere's center and the torus's generating circle's center as a variable, say D, and then find the positions where the torus and sphere are tangent along a circle.

Let me re-define the problem with variables. Let me suppose the sphere is centered at the origin, and the generating circle of the torus is centered at (D, 0, 0). The axis of the torus is the z-axis (so the generating circle is in the x-y plane). Wait, but if the generating circle is centered at (D,0,0), then the axis must be the line perpendicular to the generating circle's plane at its center. Since the generating circle is in the x-y plane, the axis would be the z-axis only if D=0. Otherwise, the axis is the line through (D,0,0) perpendicular to the x-y plane, i.e., the line x=D, y=0, z任意. So the axis is parallel to the z-axis but shifted to (D,0,0).

Therefore, the equation of the torus would be:

For any point (x,y,z), the distance to the axis (which is the line x=D, y=0, z任意) must satisfy the torus equation. The distance from (x,y,z) to the axis is sqrt( (x - D)^2 + y^2 ). Therefore, the torus equation is:

( sqrt( (x - D)^2 + y^2 ) - R )^2 + z^2 = r^2,

where R is the major radius (distance from generating circle's center to the axis, which is zero here because the generating circle is centered on the axis? Wait, no. Wait, the generating circle is in the plane perpendicular to the axis, centered at a point on the axis. Therefore, the major radius R is the distance from the center of the generating circle to the axis, which is zero? That can't be. Wait, confusion again.

Wait, let me clarify. The major radius R is the distance from the center of the generating circle to the axis of rotation. Since the generating circle is centered at (D,0,0) and the axis of rotation is the line through (D,0,0) perpendicular to the generating circle's plane (the x-y plane). Therefore, the axis is the line x=D, y=0, z任意. The distance from the generating circle's center (D,0,0) to the axis is zero, so the major radius R is zero? That can't be right.

Wait, I think I messed up the axis definition. Let me start over.

Let me define the torus such that its axis of rotation is the z-axis. Then the generating circle is in a plane perpendicular to the z-axis, say the x-y plane, centered at (R, 0, 0), where R is the major radius (6). The generating circle has radius r = 3. Therefore, the equation of the torus is:

( sqrt(x² + y²) - R )² + z² = r²,

which is the standard equation.

Now, the sphere is centered at the origin with radius 11. The problem is to find the radii r_i and r_o of the circles where the torus is externally tangent to the sphere in two different configurations.

To find these configurations, we need to consider that the torus can be positioned such that it is tangent to the sphere either "from the inside" or "from the outside". However, since the sphere is fixed at the origin and the torus is also fixed with its axis along the z-axis, perhaps the two configurations correspond to the torus being tangent along a circle in two different positions: one closer to the sphere's center and one farther away.

Alternatively, maybe the torus can be rotated or moved, but the problem states "when T rests on the outside of S" and "when T rests on the outside of S" again, which is confusing. Alternatively, perhaps the problem allows the torus to be placed such that it's either linked with the sphere or not?

Alternatively, perhaps the key is to realize that the torus has two "sides": the outer surface and the inner surface. The outer surface is farther from the center, and the inner surface is closer. The sphere can be tangent to either the outer or inner surface of the torus, leading to two different radii r_o and r_i.

Ah! That might be it. The torus has an outer radius (distance from center to the outer edge) of R + r = 6 + 3 = 9, and an inner radius of R - r = 6 - 3 = 3. So, if the sphere is tangent to the outer surface of the torus, the distance from the sphere's center to the torus's center must be such that the sphere touches the outer edge. Similarly, tangent to the inner surface would require the sphere to reach the inner edge.

Wait, but the sphere is fixed at the origin, and the torus's center (the generating circle's center) is at (6,0,0). The distance between centers is 6. The sphere has radius 11, so the distance from the sphere's center to the torus's center is 6.

If we consider the outer surface of the torus, the farthest point from the sphere's center would be along the line connecting the centers, so at (6 + 3, 0, 0) = (9,0,0). The distance from the sphere's center to this point is 9, which is less than the sphere's radius 11, so the sphere would encompass that point.

Alternatively, the closest point on the torus to the sphere's center is (6 - 3, 0, 0) = (3,0,0), at distance 3 from the origin. Since the sphere's radius is 11, which is much larger, the entire torus is inside the sphere. Therefore, to have the torus tangent to the sphere, we need to move the torus away from the sphere's center.

Wait, perhaps the problem allows the torus to be translated so that it is tangent to the sphere. The problem says "when T rests on the outside of S", so maybe the torus is placed outside the sphere, touching it. Since the sphere is larger, the torus must be placed such that it's externally tangent, meaning the sphere is inside the torus's "hole"? But the hole has a radius of 3, which is too small for the sphere of radius 11.

Alternatively, perhaps the torus is scaled? No, the problem states the torus has fixed radii.

Hmm, perhaps I need to consider that the torus can be rotated in space so that it's tangent along a different circle. Alternatively, maybe the problem is considering the torus and sphere in different orientations.

Alternatively, maybe the key is to compute the possible radii of the circles of tangency when the torus is tangent to the sphere in two different ways. Let me try to set up the equations properly.

Let me denote the sphere as S: x² + y² + z² = 11².

The torus T is given by (√(x² + y²) - 6)^2 + z² = 3².

To find their points of intersection, we can solve these equations simultaneously. Let me subtract the equations as before.

First, expand the torus equation:

(√(x² + y²) - 6)^2 + z² = 9

Expanding the square:

(x² + y²) - 12√(x² + y²) + 36 + z² = 9

So,

(x² + y² + z²) - 12√(x² + y²) + 36 = 9

But from the sphere's equation, x² + y² + z² = 121. Substitute that into the equation:

121 - 12√(x² + y²) + 36 = 9

So,

157 - 12√(x² + y²) = 9

Then,

-12√(x² + y²) = 9 - 157 = -148

Divide both sides by -12:

√(x² + y²) = 148 / 12 = 37/3 ≈ 12.333...

But √(x² + y²) is the radial distance in the x-y plane. Let me denote ρ = √(x² + y²). Then ρ = 37/3.

Then, substituting back into the sphere's equation:

ρ² + z² = 121 ⇒ z² = 121 - (37/3)^2.

Calculate (37/3)^2: 37² = 1369, so (37/3)^2 = 1369/9 ≈ 152.111...

Thus, z² = 121 - 1369/9 = (1089/9 - 1369/9) = (-280)/9.

This is negative, which is impossible. Therefore, there is no real solution, meaning the torus and sphere do not intersect when the torus is centered at (6,0,0).

This suggests that my initial assumption that the torus is centered at (6,0,0) relative to the sphere's center is incorrect. The problem must involve the torus being positioned such that it can be tangent to the sphere. Therefore, the distance between the centers is variable.

Let me denote the distance between the centers (sphere at origin, torus generating circle center at (D,0,0)) as D. Then, we can perform the same analysis with D as a variable.

Let me re-derive the equations with variable D.

The torus equation when the generating circle is at (D,0,0) with major radius R = 6 (wait, hold on, the major radius R is the distance from the generating circle's center to the axis. If the axis is the z-axis, then the major radius R is D, since the generating circle is at (D,0,0). Wait, this is crucial.

Wait, I think I made a mistake earlier. The major radius R of the torus is the distance from the center of the generating circle to the axis of rotation. If the axis of rotation is the z-axis, then the generating circle is in the plane perpendicular to the z-axis (say the x-y plane) at some point (R, 0, 0). Therefore, the major radius R is exactly the distance from (R,0,0) to the z-axis, which is R. So in the problem statement, the torus is created by revolving a circle of radius 3 around an axis in its plane at a distance 6 from the center, so the major radius R is 6. Therefore, the generating circle is at (6,0,0), and the axis is the z-axis.

Therefore, the torus equation is indeed (√(x² + y²) - 6)^2 + z² = 9.

The sphere is at the origin with radius 11. The problem is to find when the torus is tangent to the sphere along a circle, in two different configurations (probably when the torus is shifted along the line connecting their centers).

Wait, but if the torus is fixed with its generating circle at (6,0,0), then the distance between the sphere's center and the torus's generating circle center is fixed at 6. But as we saw earlier, this leads to no real intersection. Therefore, the problem must involve moving the torus so that its generating circle's center is at a different distance from the sphere's center.

Wait, perhaps the problem allows the torus to be translated so that it can be tangent to the sphere. The problem says "when T rests on the outside of S", so maybe the torus is placed such that it is tangent to the sphere, and "resting" implies that it is in contact along a circle.

Therefore, let me consider that the torus can be moved in space so that it is tangent to the sphere. Let me denote the center of the generating circle of the torus as point C, located at a distance D from the sphere's center O. The axis of the torus is still perpendicular to the line OC, so that the torus's equation can be written in a coordinate system where O is at (0,0,0), C is at (D,0,0), and the axis is the line through C perpendicular to OC (i.e., along the y-z plane at x = D).

Therefore, the equation of the torus in this coordinate system is:

( sqrt( (x - D)^2 + y^2 ) - R )^2 + z^2 = r^2,

where R is the major radius (6), and r is the minor radius (3).

The sphere's equation is x² + y² + z² = 11².

We need to find D such that these two surfaces are tangent along a circle.

To find the condition for tangency, their intersection must be a single circle (so the system has a single solution curve), and the gradients of the two surfaces must be colinear along that circle.

Alternatively, the intersection curve must have a single component (the circle) and the surfaces must touch but not cross each other there.

Let me try to find the intersection by solving the equations.

First, write the torus equation:

( sqrt( (x - D)^2 + y^2 ) - 6 )^2 + z^2 = 9.

Expand this:

( (sqrt( (x - D)^2 + y^2 ) - 6 )^2 ) + z^2 = 9,

which becomes:

( (x - D)^2 + y^2 ) - 12 sqrt( (x - D)^2 + y^2 ) + 36 + z^2 = 9,

so,

( (x - D)^2 + y^2 + z^2 ) - 12 sqrt( (x - D)^2 + y^2 ) + 36 = 9,

then,

( (x - D)^2 + y^2 + z^2 ) - 12 sqrt( (x - D)^2 + y^2 ) = -27.

Now, the sphere equation is x² + y² + z² = 121. Let me denote S = x² + y² + z² = 121.

Let me also denote ρ² = (x - D)^2 + y^2. Then the torus equation becomes:

(ρ² + z²) - 12 ρ + 36 = 9,

Wait, no. Wait, let me see:

Wait, (x - D)^2 + y^2 = ρ², so (x - D)^2 + y^2 + z^2 = ρ² + z².

But the sphere equation is x² + y² + z² = 121. Let me express (x - D)^2 + y^2 + z^2 in terms of the sphere's equation:

(x - D)^2 + y^2 + z^2 = x² - 2Dx + D² + y² + z² = (x² + y² + z²) - 2Dx + D² = 121 - 2Dx + D².

Therefore, substituting into the torus equation:

[121 - 2Dx + D²] - 12 sqrt( (x - D)^2 + y^2 ) + 36 = 9,

Simplify:

121 + D² - 2Dx + 36 - 12 sqrt( (x - D)^2 + y^2 ) = 9,

So,

D² - 2Dx + 157 - 12 sqrt( (x - D)^2 + y^2 ) = 9,

Thus,

D² - 2Dx + 148 = 12 sqrt( (x - D)^2 + y^2 ).

This is getting complicated. Maybe instead of trying to solve for x and y, I can consider the condition for the surfaces to be tangent.

The gradient of the sphere at any point (x,y,z) is proportional to (2x, 2y, 2z). The gradient of the torus equation is more complex. Let me compute it.

Let me denote the torus equation as F(x,y,z) = [sqrt( (x - D)^2 + y^2 ) - 6]^2 + z^2 - 9 = 0.

Compute the gradient ∇F:

First, let me compute partial derivatives.

Let me denote ρ = sqrt( (x - D)^2 + y^2 ), so F = (ρ - 6)^2 + z^2 - 9.

Then,

dF/dx = 2(ρ - 6)*( ( (x - D) ) / ρ ) + 0,

dF/dy = 2(ρ - 6)*( y / ρ ) + 0,

dF/dz = 2(ρ - 6)*0 + 2z = 2z.

Therefore,

∇F = [ 2(ρ - 6)*( (x - D)/ρ ), 2(ρ - 6)*( y / ρ ), 2z ]

The gradient of the sphere G(x,y,z) = x² + y² + z² - 121 is ∇G = (2x, 2y, 2z).

For the surfaces to be tangent at a point, their gradients must be colinear. Therefore, there exists a scalar λ such that:

∇F = λ ∇G.

Therefore,

2(ρ - 6)*( (x - D)/ρ ) = λ * 2x,

2(ρ - 6)*( y / ρ ) = λ * 2y,

2z = λ * 2z.

Additionally, the point (x,y,z) must lie on both surfaces:

1. (ρ - 6)^2 + z^2 = 9,

2. x² + y² + z² = 121.

Let me analyze the equations from the gradient condition.

First, from the z-component:

2z = λ * 2z ⇒ either z = 0 or λ = 1.

Case 1: z ≠ 0 ⇒ λ = 1.

Then, from the x and y components:

2(ρ - 6)*( (x - D)/ρ ) = 2x,

2(ρ - 6)*( y / ρ ) = 2y.

Divide both equations by 2:

(ρ - 6)*( (x - D)/ρ ) = x,

(ρ - 6)*( y / ρ ) = y.

Let me rearrange the first equation:

(ρ - 6)(x - D)/ρ - x = 0,

Multiply through by ρ:

(ρ - 6)(x - D) - x ρ = 0,

Expand:

ρ(x - D) -6(x - D) - x ρ = 0,

ρ x - ρ D -6x + 6D - x ρ = 0,

The ρ x terms cancel:

-ρ D -6x + 6D = 0,

Rearranged:

-ρ D -6x = -6D,

Multiply both sides by -1:

ρ D + 6x = 6D,

Thus,

ρ D = 6D -6x,

Divide both sides by D (assuming D ≠ 0):

ρ + (6x)/D = 6,

Wait, no, let me see:

Wait, ρ D + 6x = 6D ⇒ ρ D = 6D -6x ⇒ ρ = 6 - (6x)/D.

Similarly, from the second equation:

(ρ -6)(y)/ρ = y,

Assuming y ≠ 0, we can divide both sides by y:

(ρ -6)/ρ = 1,

Thus,

ρ -6 = ρ ⇒ -6 = 0,

Which is impossible. Therefore, the assumption y ≠ 0 leads to a contradiction. Therefore, either y = 0 or our assumption λ =1 is only valid when y=0.

Therefore, in case 1 (z ≠0 and λ=1), we must have y =0.

So, with y =0, let's proceed.

So, y =0, z ≠0, and λ=1.

From the first equation (with y=0):

ρ = sqrt( (x - D)^2 + 0 ) = |x - D|.

Since ρ is a distance, it's positive.

So, ρ = |x - D|. Let me write ρ = |x - D|. Since we can have x < D or x > D, but let's keep it as is for now.

From the first equation's result:

ρ D +6x =6D,

Substituting ρ = |x - D|,

|x - D| * D +6x =6D.

Let me consider two subcases based on the sign of (x - D).

Subcase 1a: x ≥ D ⇒ ρ = x - D.

Then,

(x - D)*D +6x =6D,

Expand:

Dx - D² +6x =6D,

x(D +6) =6D + D²,

Thus,

x = (6D + D²)/(D +6) = D(D +6)/(D +6) )= D.

Therefore, x = D. But if x = D, then ρ = 0, which would make the first equation:

0 * D +6*D =6D ⇒ 6D =6D, which holds.

But if x = D, then from the sphere equation with y=0 and z ≠0:

x² + 0 + z² =121 ⇒ D² + z² =121,

And from the torus equation:

(ρ -6)^2 + z² =9 ⇒ (0 -6)^2 + z² = 9 ⇒ 36 + z² =9 ⇒ z² = -27,

Which is impossible. Therefore, Subcase 1a leads to no solution.

Subcase 1b: x < D ⇒ ρ = D -x.

Then,

(D -x)*D +6x =6D,

Expand:

D² - Dx +6x =6D,

D² +x(-D +6) =6D,

Solve for x:

x(-D +6) =6D - D²,

x = (6D - D²)/( -D +6 ) = [ -D(D -6) ] / [ - (D -6) ] = D.

Again, x = D, which contradicts x < D. Therefore, no solution in this subcase either.

Therefore, Case 1 (z ≠0) leads to no solution.

Case 2: z =0.

Then, from the gradient equations, the z-component equation is satisfied for any λ, since both gradients have z-component 0 (since z=0).

Now, with z=0, the equations simplify.

First, the sphere equation becomes x² + y² = 121.

The torus equation becomes:

( sqrt( (x - D)^2 + y^2 ) -6 )^2 =9,

Taking square roots,

sqrt( (x - D)^2 + y^2 ) -6 = ±3,

Thus,

sqrt( (x - D)^2 + y^2 ) = 6 ±3,

So either sqrt(...) =9 or sqrt(...) =3.

Therefore, the intersection when z=0 consists of two circles:

1. (x - D)^2 + y^2 = 81 (radius 9 circle centered at (D,0))

2. (x - D)^2 + y^2 = 9 (radius 3 circle centered at (D,0))

These must coincide with the sphere's intersection (x² + y² =121) for tangency.

For the surfaces to be tangent along a circle, this circle must lie in the plane z=0 and be a single circle, so one of the two circles from the torus must coincide with the sphere's intersection.

Therefore, either:

1. (x - D)^2 + y^2 =81 and x² + y² =121 must be the same circle, or

2. (x - D)^2 + y^2 =9 and x² + y² =121 must be the same circle.

But for two circles to be the same, their centers and radii must coincide.

First, consider case 2a: (x - D)^2 + y^2 =81 and x² + y² =121.

Expanding the first equation:

x² - 2Dx + D² + y² =81,

But x² + y² =121, so substitute:

121 -2Dx + D² =81,

Thus,

-2Dx + D² = -40,

=> 2Dx = D² +40,

=> x = (D² +40)/(2D).

But this must hold for all points on the circle, which is only possible if the coefficients of x are zero unless it's a single point. Since this is supposed to be a circle, this can only happen if the linear term in x is zero, which requires D=0, but then:

If D=0, then the equation becomes x² + y² =81 and x² + y²=121, which can't be the same. Therefore, this case is impossible unless the circles are the same, which would require D=0 and 81=121, which is false.

Similarly, case 2b: (x - D)^2 + y^2 =9 and x² + y² =121.

Expanding:

x² -2Dx + D² + y² =9,

Again, substituting x² + y² =121,

121 -2Dx + D² =9,

Thus,

-2Dx + D² = -112,

2Dx = D² +112,

x = (D² +112)/(2D).

Again, this must hold for all points on the circle, which requires that the linear term in x is zero, so D must be zero, but then:

If D=0, then x² + y² =9 and x² + y²=121, which is impossible.

Therefore, the only way for the circles to coincide is if the equations are identical, which requires that the centers and radii match.

Alternatively, perhaps the intersection is a single point? Wait, but the problem states that they are tangent along a circle, so the intersection must be a circle.

Alternatively, maybe the tangency occurs along the circle where z=0, but the gradients are colinear. Let me check the gradient condition for z=0.

At z=0, the gradient of the sphere is (2x, 2y, 0).

The gradient of the torus F at z=0 is:

∇F = [ 2(ρ -6)*( (x - D)/ρ ), 2(ρ -6)*( y / ρ ), 0 ]

So, for gradients to be colinear, there must exist λ such that:

2(ρ -6)*( (x - D)/ρ ) = λ * 2x,

2(ρ -6)*( y / ρ ) = λ * 2y,

0 = λ *0.

The third equation is satisfied.

Assuming y ≠0, we can divide the second equation by 2y:

(ρ -6)/ρ = λ,

Similarly, first equation:

(ρ -6)(x - D)/ρ = λ x,

Substituting λ from the second equation:

(ρ -6)(x - D)/ρ = ( (ρ -6)/ρ ) x,

Multiply both sides by ρ:

(ρ -6)(x - D) = (ρ -6)x,

Assuming ρ ≠6 (otherwise LHS and RHS would be zero, but let's check):

If ρ ≠6, we can divide both sides by (ρ -6):

x - D = x ⇒ -D =0 ⇒ D=0,

But D is the distance from the sphere's center to the torus's generating circle center, so D=0 would mean they are at the same center. Let's see if that works.

If D=0, then the torus is centered at the origin, so its equation is (sqrt(x² + y²) -6)^2 + z² =9. The sphere is x² + y² + z² =121.

The intersection when z=0 would be (sqrt(x² + y²) -6)^2 =9 ⇒ sqrt(x² + y²)=9 or 3. So circles of radius 9 and 3. The sphere at z=0 has radius sqrt(121)=11. So the circle of radius 9 is inside, and 3 is also inside. So they are not tangent.

Therefore, the only solution is when ρ=6.

If ρ=6, then from the gradient equations:

λ = (6 -6)/6 =0,

So gradients would be zero? Wait, if ρ=6, then the first component of ∇F is 2(0)*(...)=0, similarly the second component is 0, so ∇F is (0,0,0), which is not possible.

Therefore, the only solution is when y=0. Let me consider y=0.

If y=0, then the gradient condition's second equation is satisfied for any λ.

From the first equation:

(ρ -6)*( (x - D)/ρ ) = λ x,

With ρ = |x - D| (since y=0).

Let me write ρ = |x - D|. Let me consider ρ = x - D if x ≥ D, else D -x.

But let's proceed with ρ = sqrt( (x - D)^2 ) = |x - D|.

Let me suppose ρ ≠0 (so x ≠ D).

Then,

( (|x - D| -6 ) * ( (x - D)/|x - D| ) ) = λ x,

The term (x - D)/|x - D| is the sign of (x - D). Let me denote s = sign(x - D). Then,

( s|x - D| -6 ) * s = λ x,

=> ( |x - D| s -6s ) = λ x,

But |x - D| s = x - D,

Therefore,

(x - D) -6s = λ x,

Thus,

x - D -6s = λ x,

Rearranged,

x(1 - λ) = D +6s,

So,

x = (D +6s)/(1 - λ).

This is getting too convoluted. Maybe it's better to consider specific configurations where the torus is tangent to the sphere along a circle in the plane z=0.

Suppose the circle of tangency lies in the plane z=0. Then, the intersection of the torus and sphere in this plane must be a single circle, and the gradients must be colinear there.

From earlier, in the plane z=0, the torus has two circles: (x - D)^2 + y^2 =81 and =9. The sphere has x² + y²=121.

For these to be tangent, one of the torus's circles must be tangent to the sphere's circle.

The sphere's circle has radius 11, centered at (0,0). The torus's circles are centered at (D,0) with radii 9 and 3.

The distance between the centers is D.

For the circle of radius 9: the distance between centers is D. For them to be tangent, the distance between centers must be equal to 11 ±9.

So,

Either D = 11 +9 =20, or D=11 -9=2.

Similarly for the smaller circle (radius 3):

Distance between centers D must be 11 ±3 ⇒ D=14 or D=8.

Therefore, possible D values are 2, 8,14,20.

But the torus's generating circle is at distance D from the sphere's center. The major radius of the torus is 6, so the generating circle is part of the torus.

Wait, but the problem states that the torus has a major radius of 6 (distance from generating circle's center to the axis). If the axis is the z-axis, then D is the distance from the sphere's center to the generating circle's center. The major radius is fixed at 6, so the axis must be the z-axis, so D is variable depending on the position.

Wait, but in the problem statement, the torus is defined with major radius 6, so the generating circle is always 6 units from the axis (the z-axis). Therefore, the generating circle's center is at (D,0,0), so the distance from the axis (z-axis) is D. Therefore, the major radius R is D. Wait, this is a critical point.

Wait, I think I made a mistake earlier. The major radius R of the torus is the distance from the generating circle's center to the axis of rotation. If the axis is the z-axis, then the generating circle's center is at (R,0,0). Therefore, R is fixed at 6, so D must be 6. Therefore, the generating circle is fixed at (6,0,0).

This brings us back to the original setup where D=6. But earlier, we saw that when D=6, the equations lead to a contradiction, implying no real intersection.

This suggests that the problem might require considering that the torus can be rotated or the axis can be oriented differently, but the problem states "when T rests on the outside of S", implying that the torus is placed in contact with the sphere, so perhaps the axis is not fixed along the z-axis but can be oriented such that the torus is tangent.

Alternatively, maybe the problem is considering the torus in a different orientation where the axis is not along the line connecting the centers.

Alternatively, perhaps the key is to realize that the radii r_i and r_o correspond to the two possible circles of intersection when the torus is tangent to the sphere, and their difference can be found via geometric considerations without needing to go through the algebra.

Let me think of the torus as a surface with two important radii: the major radius R=6 and minor radius r=3. The sphere has radius 11.

When the torus is tangent to the sphere externally, the distance between their centers must satisfy certain conditions. Let me denote the distance between the centers as d.

For the torus and sphere to be tangent, the distance between centers d must satisfy:

For external tangency, the distance between centers is equal to the sum of their "radii" in some sense. However, the torus is a more complex shape.

Alternatively, consider the closest and farthest points on the torus from the sphere's center. The closest point on the torus to the sphere's center (origin) is along the line connecting the centers, at a distance of D - (R + r) = 6 - (6 +3)= -3, which doesn't make sense. Wait, no.

Wait, the closest point on the torus to the origin would be along the line from the origin to the generating circle's center (6,0,0), towards the origin. The generating circle is centered at (6,0,0) with radius 3, so the closest point on the generating circle is at (6 - 3, 0,0) = (3,0,0). Therefore, the closest point on the torus is 3 units away from the origin. The farthest point would be (6 +3,0,0) = (9,0,0), 9 units away.

Since the sphere has radius 11, the sphere engulfs the entire torus, so they don't touch. Therefore, to have them tangent, the torus must be moved away from the sphere's center.

Suppose we move the torus along the line connecting their centers so that the distance between the centers is now D. The generating circle's center is at distance D from the origin. The closest point on the torus to the origin is then D - (R + r) = D -9, and the farthest is D +9.

To have the torus tangent to the sphere of radius 11, either:

1. The closest point on the torus is at distance 11 from the origin: D -9 =11 ⇒ D=20,

or

2. The farthest point on the torus is at distance 11 from the origin: D +9 =11 ⇒ D=2.

These are the two possible positions where the torus is tangent to the sphere along a single point. However, the problem states that they are tangent along a circle, so perhaps when the distance D is such that the sphere and torus intersect in a circle.

Wait, but tangency along a circle would require that the distance between centers is such that the sphere touches the torus along a circle, which occurs when the sphere's surface is tangent to the torus's surface along a circle. This would happen when the sphere's radius satisfies a certain condition relative to the torus's geometry.

Let me consider the cross-section again. In the plane containing the sphere's center and the torus's generating circle center (the x-axis), the torus appears as two circles: the generating circle at (D,0) with radius 3, and its mirror image. The sphere appears as a circle of radius 11.

The points of tangency in this cross-section would be points where the sphere's circle and the generating circle are tangent. The distance between the centers is D.

For the circles to be tangent, the distance between centers must be equal to the sum or difference of their radii.

So, either:

D = 11 + 3 =14 (externally tangent),

or D = 11 -3 =8 (internally tangent).

Therefore, when D=14, the generating circle is outside the sphere, tangent at one point. When D=8, it's inside, tangent at one point.

However, in 3D, the tangency in cross-section would correspond to a circle of tangency in 3D. Therefore, the radii r_i and r_o correspond to the radii of these circles when D=8 and D=14.

Wait, but how do we find the radius of the circle of tangency?

The circle of tangency lies in a plane perpendicular to the line connecting the centers (the x-axis). The radius of this circle can be found using the Pythagorean theorem.

Let me consider when D=8 (internally tangent). The sphere's radius is 11, the distance between centers is 8. The radius of the circle of tangency would be sqrt(11² -8²)=sqrt(121-64)=sqrt(57).

Similarly, when D=14 (externally tangent), the radius is sqrt(11² -14²)=sqrt(121-196)=sqrt(-75), which is imaginary. Wait, that can't be.

Wait, no. Wait, when D=14, the distance between centers is 14, which is greater than the sphere's radius 11, so the sphere is inside the generating circle? Wait, no. Let me think again.

Wait, in cross-section, the generating circle has radius 3 and is centered at D=14 from the sphere's center. The sphere has radius 11. The distance between centers is 14, so the sphere and the generating circle are externally tangent when D=14 (since 14 =11 +3). The point of tangency is along the line connecting the centers, so in 3D, this corresponds to a circle in the plane perpendicular to that line at the point of tangency.

The radius of the circle of tangency would be the radius of the sphere's cross-section at that plane.

Wait, let me formalize this.

When the generating circle and the sphere's cross-section are tangent at a point, the circle of tangency in 3D is the set of points on both the sphere and torus that lie in the plane perpendicular to the line connecting the centers at the point of tangency.

The radius of this circle can be calculated as follows:

Suppose the distance between the centers is D. The point of tangency in the cross-section is along the line connecting the centers, at a distance of 11 from the sphere's center and 3 from the generating circle's center.

Therefore, the coordinates of the tangency point in cross-section are:

For D =8 (internally tangent):

The point is 11 - 8 =3 units from the generating circle's center towards the sphere's center. Wait, no:

Wait, if D=8, the generating circle is at distance 8 from the sphere's center. The sphere's radius is 11, so the point on the sphere towards the generating circle is at distance 11 from the origin, so the point is along the line towards the generating circle's center (at D=8), so the coordinates are ( (11/8)*8, 0 )? Wait, no.

Wait, the point of tangency is along the line connecting the centers. Let me denote the sphere's center as O and the generating circle's center as C. The point of tangency P lies along the line OC.

For internal tangency (D =8):

The distance from O to P is 11 (sphere's radius), and from C to P is 3 (generating circle's radius). Since C is 8 units from O, then OP = OC - CP ⇒ 11 =8 -3? No, that gives 11=5, which is wrong.

Wait, actually, for internal tangency, the point P is between O and C. So:

OP + PC = OC,

But OP =11, PC=3, OC= D=8,

Thus, 11 +3 =8? 14=8? No, that's impossible.

Wait, perhaps I have it backwards. For internal tangency, the generating circle is inside the sphere, so the point P is in the direction from C away from O.

Wait, let me think carefully.

If the generating circle is at D=8 from O, and has radius 3, then the closest point on the generating circle to O is at distance D -3 =5, and the farthest is D +3=11.

Therefore, the farthest point is exactly at distance 11 from O, which is on the sphere. Therefore, the point (D +3) along the line OC is at distance 8 +3=11 from O, so that point is on both the generating circle and the sphere. Therefore, the point of tangency is that farthest point.

Therefore, the tangency occurs at a single point in cross-section, but in 3D, this corresponds to a circle in the plane perpendicular to OC at that point.

The radius of this circle can be found by considering the sphere's radius at that plane.

The distance from the sphere's center to the plane is 11 (since the point is on the sphere's surface), but wait, the plane is at distance 11 from the center, so the radius of the circle would be sqrt(11² -11²)=0, which is just a point. That can't be right.

Hmm, confusion arises because in 3D, the tangency along a single point in cross-section implies that the circle of tangency has zero radius, which contradicts the problem's statement.

Alternatively, perhaps the circle of tangency is not in the cross-sectional plane. Let me consider a different approach.

The circle of tangency lies in a plane perpendicular to the line connecting the centers of the sphere and the generating circle. Let me denote this line as the x-axis, with the sphere at O(0,0,0) and the generating circle's center at C(D,0,0). The plane of tangency is perpendicular to the x-axis at some point P along OC.

In this plane, the sphere's intersection is a circle of radius sqrt(11² - p²), where p is the distance from O to the plane. Similarly, the torus's intersection in this plane is a circle (or two circles).

The torus's equation in this plane (x = p) is:

( sqrt( (p - D)^2 + y^2 ) -6 )^2 + z^2 =9.

But in the plane x=p, the coordinates are (p, y, z). The sphere's equation gives y² + z² =11² -p².

The torus's equation can be written as:

( sqrt( (p - D)^2 + y^2 ) -6 )^2 + z^2 =9.

To have tangency, these two equations must have a common solution with the gradients colinear.

Alternatively, for the circles to be tangent, the distance between their centers must equal the sum or difference of their radii.

The sphere's circle in the plane x=p has center (p,0,0) [projection onto the plane] and radius sqrt(11² - p²).

The torus's intersection in this plane would be more complex, but perhaps in the plane of tangency, the torus's intersection is a circle with radius related to its minor radius.

Alternatively, perhaps the radius of the circle of tangency is the minor radius of the torus, but scaled.

Wait, when the torus is tangent to the sphere, the circle of tangency is a "latitude" on the torus. The radius of this circle can be found using the geometry of the torus.

The torus has major radius R=6 and minor radius r=3. A circle on the torus in the plane at a distance d from the z-axis (the axis of the torus) has radius 2πr, but I need to think differently.

Wait, the radius of a circular cross-section of the torus in a plane perpendicular to its axis (the z-axis) at a distance x from the center is 2r sin(theta), but perhaps I need to recall the formula for the radius of a parallel on the torus.

The radius of a circle (parallel) on the torus at a distance θ from the axis is given by r_parallel = r * sinθ, but I might be misremembering.

Alternatively, parametrizing the torus:

A point on the torus can be parametrized as:

x = (R + r cos φ) cos θ,

y = (R + r cos φ) sin θ,

z = r sin φ,

where θ is the angle around the torus's axis, and φ is the angle around the generating circle.

The radius of the circle at a fixed θ (a "latitude") is r, but projected onto the plane perpendicular to the axis. Wait, perhaps the radius in the plane perpendicular to the axis at a point is r.

Alternatively, the radius of the circle of tangency would be the minor radius of the torus, 3, but adjusted based on the distance from the sphere's center.

This is getting too vague. Let me try to use the earlier cross-sectional analysis.

When the generating circle is at distance D from the sphere's center, the point of tangency in cross-section is at distance 11 from the sphere's center and 3 from the generating circle's center.

Thus, the coordinates of the tangency point P are along the line connecting O and C, at a distance of 11 from O and 3 from C.

Therefore, the distance between O and C is D = OP + PC =11 +3=14 (external tangent) or D= |11 -3|=8 (internal tangent).

Therefore, the two possible distances between centers are D=14 and D=8.

Now, the circle of tangency lies in the plane perpendicular to OC at point P. The radius of this circle can be found using the sphere's radius and the distance from the sphere's center to the plane.

The distance from O to the plane is the distance from O to P, which is 11 for external tangent (D=14) and 8 for internal tangent (D=8)? Wait, no.

Wait, for external tangent (D=14), the point P is 11 units from O and 3 units from C (since D=14=11+3). The plane is perpendicular to OC at P, so the distance from O to the plane is 11. The radius of the sphere's intersection with the plane is sqrt(11² -11²)=0, which is just a point. That can't be.

Wait, this suggests that the tangency is only at a single point, which contradicts the problem's statement of a circle.

Similarly, for internal tangent (D=8), the point P is 11 units from O (since it's on the sphere) and 3 units from C (since it's on the generating circle). The distance from O to C is 8, so the point P is located 11 units from O towards C, but since OC is only 8 units, this is impossible. Wait, no:

Wait, if D=8, then the generating circle is 8 units from O. The point P is on both the sphere and the generating circle. The distance from O to P is 11, but the generating circle is only 8 units away, so the point P must be on the opposite side of C from O.

Wait, let me compute coordinates.

Let me set O at (0,0,0), C at (D,0,0). The generating circle is centered at C with radius 3. The sphere is centered at O with radius 11.

For external tangent (D=14):

The point P is along the line OC, beyond C from O. So coordinates of P are (D +3,0,0) = (17,0,0). The distance from O is 17, which is greater than 11, so this is outside the sphere. Not on the sphere.

Wait, this is confusing. Let me recast:

For external tangent between the generating circle (radius 3) and the sphere (radius 11), the distance between centers must be D =11 +3=14. The point of tangency is located at O + (11/D)*(C - O) = ( (11/14)*14,0,0 )=(11,0,0). Wait, no:

Wait, the point of tangency P lies along the line connecting O and C. Let me denote vector OC as pointing from O to C. The point P is located at a distance of 11 from O towards C, but since the generating circle is at D=14, which is farther than 11, the point P is between O and C.

Wait, coordinates:

O is at (0,0,0), C is at (14,0,0). The generating circle has radius 3, so the point closest to O on the generating circle is at (14 -3,0,0)=(11,0,0). This point is exactly 11 units from O, so it lies on the sphere. Therefore, the point (11,0,0) is the point of tangency.

Similarly, for internal tangent (D=8):

The generating circle is at (8,0,0). The point farthest from O on the generating circle is at (8 +3,0,0)=(11,0,0), which is on the sphere. So the point (11,0,0) is again the point of tangency.

Wait, so in both cases, the point of tangency is (11,0,0). That can't be right.

Wait, no. For internal tangent (D=8), the generating circle is inside the sphere. The point of tangency would be in the direction away from O.

Wait, let me compute:

For D=8, the generating circle is centered at (8,0,0). The point on the generating circle farthest from O is (8 +3,0,0)=(11,0,0), which is on the sphere. The point closest is (5,0,0).

Therefore, the point (11,0,0) is the tangency point for both external and internal tangent? That seems odd.

Wait, perhaps the external tangent is when the generating circle is outside the sphere, so D=14, and the point of tangency is (11,0,0). The internal tangent is when the generating circle is inside the sphere, D=8, and the point of tangency is also (11,0,0).

But in both cases, the tangency is at the same point? That doesn't make sense.

Alternatively, perhaps for internal tangent, the point is on the opposite side.

Wait, if D=8, the generating circle is at (8,0,0). The sphere's surface in the opposite direction (away from the generating circle) would be at (-11,0,0). The generating circle can't reach there.

Alternatively, maybe the internal tangent is when the generating circle is inside the sphere, and the point of tangency is on the side towards O.

The closest point on the generating circle to O is (5,0,0). The distance from O is 5, which is less than 11, so not on the sphere.

Hmm, this is perplexing. It seems that when D=14, the generating circle touches the sphere at (11,0,0), and when D=8, the generating circle also touches the sphere at (11,0,0). That suggests that both configurations result in the same tangency point, which contradicts the problem's mention of two different radii r_i and r_o.

Therefore, my approach must be flawed. Perhaps the circle of tangency is not in the cross-sectional plane but in another plane.

Let me consider that the circle of tangency is not along the line connecting the centers but in a different orientation.

Suppose the torus is positioned such that its axis is not aligned with the line connecting the sphere's center. However, the problem states "when T rests on the outside of S", implying that the axis is aligned for stability.

Alternatively, the radius of the circle of tangency is the minor radius of the torus, 3, but scaled by some factor based on the distance D.

Wait, the minor radius is 3, so the circle of tangency could have radius 3. But the problem states that the difference r_i - r_o is a fraction, so likely not.

Alternatively, using the formula for the radius of a circle on a torus:

The radius of a circle on the torus at a distance x from the center along the axis is given by sqrt(r^2 - (R -x)^2). Wait, not sure.

Alternatively, parametrizing the torus:

A point on the torus can be written in cylindrical coordinates (ρ, θ, z), where ρ = R + r cos φ, and z = r sin φ. The radius of the circle at a fixed φ is ρ, but this might not help.

Alternatively, the radius of the circle of tangency can be found by considering the sphere and torus equations.

Let me consider the case when D=8 (internally tangent). The distance between centers is 8. The circle of tangency lies in a plane perpendicular to the line connecting the centers at the point P where they touch.

The coordinates of P are (11,0,0) as before. The plane perpendicular to OC at P is the plane x=11.

In this plane, the sphere's cross-section is a circle of radius sqrt(11² -11²)=0, which is just a point. The torus's cross-section in this plane is given by substituting x=11 into the torus equation:

( sqrt( (11 -8)^2 + y^2 ) -6 )^2 + z^2 =9,

Wait, D=8, so the generating circle is at (8,0,0).

Wait, the torus equation when the generating circle is at (D,0,0) is:

( sqrt( (x - D)^2 + y^2 ) -6 )^2 + z^2 =9.

So substituting x=11, D=8:

sqrt( (11-8)^2 + y^2 ) = sqrt(9 + y^2),

Thus,

( sqrt(9 + y^2 ) -6 )^2 + z^2 =9.

Expanding:

(9 + y^2 -12 sqrt(9 + y^2 ) +36 ) + z^2 =9,

So,

y^2 + z^2 +45 -12 sqrt(9 + y^2 ) =9,

y^2 + z^2 -12 sqrt(9 + y^2 ) +36 =0.

But in the plane x=11, the sphere's equation gives y² + z² =121 -121=0, so y=0, z=0. Thus, only the point (11,0,0) is common, so the intersection is just a point.

Therefore, the tangency is at a single point, not a circle. This contradicts the problem's statement.

This suggests that my entire approach is incorrect. Perhaps I need to consider that the torus is not translated along the line connecting the centers but rotated such that its axis is inclined.

Alternatively, the problem might involve the torus being tangent to the sphere along a circle that is not in the cross-sectional plane.

Let me try a different method using the concept of the radius of curvature.

The torus has two principal radii of curvature: the major radius R=6 and minor radius r=3. The sphere has radius 11.

When the torus is externally tangent to the sphere, the center of curvature of the torus's surface must coincide with the sphere's center.

The radius of curvature of the torus at a point depends on the direction. For the outer surface, the radius of curvature is R + r =9, and for the inner surface, it's R - r =3.

Therefore, to be tangent to the sphere of radius 11, the distance from the sphere's center to the torus's center must be 11 ± radius of curvature.

For the outer surface (radius of curvature 9):

Distance between centers =11 -9=2 or 11 +9=20.

For the inner surface (radius of curvature 3):

Distance between centers=11 -3=8 or 11 +3=14.

Therefore, the possible distances are 2,8,14,20.

The radii of the circles of tangency would correspond to the minor radius of the torus scaled by the angle.

Wait, the radius of the circle of tangency on the torus would be the minor radius r=3. But since the torus is at a distance from the sphere's center, perhaps the radius in 3D is different.

Alternatively, the radius of the circle of tangency can be found using the sphere's radius and the distance between centers.

Using the formula for the radius of a circle formed by the intersection of two spheres:

If two spheres with radii R and r are separated by distance d, the radius of their intersection circle is:

sqrt( [ (-d + R + r)(d + R - r)(d - R + r)(d + R + r) ] / (4d²) )

But in this case, one is a torus, so this formula doesn't apply.

Alternatively, when the torus is tangent to the sphere along a circle, the radius of that circle can be found by considering the minor radius of the torus and the geometry of the position.

Suppose the torus is positioned such that its center (the generating circle's center) is at distance D from the sphere's center. The circle of tangency lies in a plane perpendicular to the line connecting the centers.

The radius of the circle on the torus is its minor radius, 3. However, due to the distance D, the actual radius in 3D space would be affected by the angle.

Alternatively, the radius of the circle of tangency on the sphere is sqrt(11² - D²).

Wait, the sphere's cross-section in the plane perpendicular to the line of centers at distance p from the center has radius sqrt(11² - p²).

If the circle of tangency is in such a plane at distance p from the sphere's center, then the radius is sqrt(11² - p²).

Similarly, the torus's cross-section in that plane must also form a circle of the same radius.

The torus's cross-section in that plane would depend on its position.

Assuming the torus's generating circle is at distance D from the sphere's center, the distance from the generating circle's center to the plane is |D - p|. The radius of the torus's cross-section in that plane would be sqrt( (distance from generating circle's center)^2 + (minor radius)^2 )? Not sure.

Alternatively, the radius of the torus's cross-section in the plane is given by its minor radius, but adjusted by the angle.

This is getting too vague. Perhaps I should look for an example with D=8 and D=14 and compute the radii.

Let me try D=8 (internally tangent):

The distance between centers is 8. The circle of tangency is in the plane perpendicular to the line connecting the centers at the point where they touch.

The point of tangency is at distance 11 from the sphere's center and 3 from the generating circle's center.

Thus, the coordinates are (11,0,0). The plane is perpendicular to the x-axis at x=11.

In this plane, the sphere's radius is 0 (as before), so no circle.

Alternatively, maybe the circle of tangency is not at the point of closest approach but somewhere else.

Suppose the torus is positioned at D= sqrt(11² -3²)=sqrt(121-9)=sqrt(112)=4*sqrt(7)≈10.583. Then the circle of tangency would have radius 3. But I need to think differently.

Alternatively, using the method of Lagrange multipliers to find the extrema.

We need to find points where the gradient of the sphere and torus are colinear, which we attempted earlier.

Let me try to proceed with the equations.

We have the sphere: x² + y² + z² =121,

and the torus: (sqrt( (x - D)^2 + y^2 ) -6)^2 + z² =9.

Subtracting the equations as before:

[sphere] - [torus] = 121 - [ ( (x-D)^2 + y^2 -12 sqrt(...) +36 ) + z² ].

Wait, this might not help.

Alternatively, let me assume that the circle of tangency lies in the plane z=0.

Then, the radius of the circle is sqrt(x² + y²).

From the sphere: x² + y² =121 - z² =121.

From the torus: (sqrt( (x-D)^2 + y^2 ) -6)^2 =9 - z² =9.

Thus,

sqrt( (x-D)^2 + y^2 ) =6 ±3,

so,

(x-D)^2 + y^2 = (9)^2 or (3)^2.

Therefore, the intersection in the z=0 plane are the circles (x-D)^2 + y^2 =81 or 9, and the sphere's circle x² + y²=121.

For these to be tangent, one of the torus's circles must be tangent to the sphere's circle.

The distance between centers is D.

For the circle of radius 9:

To be tangent to the sphere's radius 11, the distance between centers must be 11 ±9.

Thus, D=11+9=20 or D=11-9=2.

Similarly for the circle of radius 3: D=11±3 ⇒14 or8.

Therefore, when D=2, the torus's outer circle (radius9) is internally tangent to the sphere (distance between centers 2=11-9). The circle of tangency has radius sqrt(11² -2²)=sqrt(121-4)=sqrt(117)=3*sqrt(13).

When D=20, the outer circle is externally tangent, radius sqrt(11² -20²) which is imaginary.

Wait, no. For D=20, the distance between centers is 20, so the radius of the sphere's cross-section at the plane of the torus's circle (which is at distance 20 from the center) would be sqrt(11² -20²), which is imaginary. So that's not possible.

Wait, perhaps the radius is calculated differently.

If the torus's generating circle is at D=2, then the circle of tangency (the sphere's intersection) has radius sqrt(11² - D²)=sqrt(121-4)=sqrt(117).

Similarly, when D=8, the inner circle of the torus (radius3) is tangent, so the radius would be sqrt(11² -8²)=sqrt(121-64)=sqrt(57).

When D=14, the inner circle would be sqrt(11² -14²)=imaginary.

Thus, the valid radii are sqrt(117) and sqrt(57).

Therefore, the two radii are r_i= sqrt(117) and r_o= sqrt(57), or vice versa.

The problem states "when T rests on the outside of S", which would correspond to the smaller distance D=2 (since the torus is closer, "resting on" the sphere from the inside?), giving a larger radius sqrt(117), and when it's on the outside (farther away at D=8?), but this is confusing.

Wait, when D=2, the generating circle is very close to the sphere's center, so the torus is mostly inside the sphere, touching it at a circle of radius sqrt(117).

When D=8, the generating circle is farther out, and the torus touches the sphere at a smaller circle sqrt(57).

Therefore, the difference would be sqrt(117) - sqrt(57). But the problem states the difference is r_i - r_o, which could be either sign.

Wait, the problem says "when T rests on the outside of S", which might correspond to D=2 (inside the sphere) giving a larger radius, and "rests on the inside" (D=8) giving a smaller radius. Thus, r_i (inside) is sqrt(117), r_o (outside) is sqrt(57), so difference is sqrt(117)-sqrt(57). But this needs to be a fraction m/n.

Hmm, but sqrt(117) and sqrt(57) are irrational. This suggests my approach is wrong.

Alternatively, perhaps the radii are 9 and 3, but that difference is 6, which is 6/1, so m+n=7, but the problem states it's a fraction.

Alternatively, perhaps the radii are 9 and 3, but scaled by some factor.

Wait, going back to the cross-section approach, when D=2, the generating circle is at 2 units from the sphere's center. The circle of tangency in the plane perpendicular to OC at the point of tangency would have radius sqrt(11² -2²)=sqrt(117).

Similarly, when D=8, it's sqrt(57).

The problem states the difference is m/n. So sqrt(117) - sqrt(57) must be rational?

But sqrt(117)=3*sqrt(13), sqrt(57)=sqrt(57). These are both irrational and likely not rational multiples.

This suggests my method is incorrect.

Perhaps I need to consider that the radius of the circle of tangency on the torus is its minor radius, 3, but adjusted by the angle between the axis and the line connecting the centers.

Let me consider the angle θ between the torus's axis and the line connecting the centers.

The radius of the circle of tangency would be r / sinθ, where r is the minor radius.

Alternatively, using similar triangles.

Alternatively, the radius can be found via the formula for the radius of a circle on a torus:

The radius of a circle on the torus at a distance x from the center is sqrt(r^2 - (R -x)^2).

Wait, perhaps the radius of the circle of tangency is given by the minor radius divided by the cosine of the angle between the axis and the line connecting the centers.

This is getting too vague.

Alternatively, let me consider the case when the torus is tangent to the sphere. The circle of tangency lies in a plane where the normal vectors of both surfaces are parallel.

The radius of the circle can be found by the following method:

The radius of the circle on the sphere is sqrt(R² - d²), where R is the sphere's radius and d is the distance from the sphere's center to the plane.

Similarly, the radius on the torus would depend on its geometry.

Assuming that the plane is at distance p from the sphere's center, then the radius on the sphere is sqrt(121 - p²).

For the torus, the radius in that plane would be its minor radius, 3, but scaled by the angle.

Alternatively, if the torus's generating circle is at distance D from the sphere's center, then the distance from the generating circle's center to the plane is |D - p|. The radius of the torus's cross-section in that plane is sqrt( (distance)^2 + (minor radius)^2 ).

Wait, no. The radius of the torus in the plane would be the minor radius, but depending on the orientation.

Alternatively, the radius of the torus's cross-section in the plane is the minor radius, 3.

Therefore, setting the two radii equal (since they must match at the circle of tangency):

sqrt(121 - p²) =3,

Thus,

121 -p²=9 ⇒ p²=112 ⇒ p= sqrt(112)=4*sqrt(7).

The distance from the sphere's center to the plane is 4√7.

The distance between the generating circle's center and the plane is |D - p|.

The radius of the torus's cross-section in the plane is 3, so:

sqrt( (D - p)^2 + 3^2 ) = sqrt( (D - p)^2 +9 ) must equal the sphere's radius in the plane, which is 3.

Wait, that would require (D - p)^2 +9 =9 ⇒ D=p.

Thus, D must equal p=4√7≈10.666.

But then the distance between centers D=4√7, and the generating circle's center is at that distance.

The radius of the circle of tangency is 3.

But this gives only one radius.

Alternatively, there are two solutions depending on the side of the plane.

If the generating circle is on the opposite side, D + p = distance, leading to D= -4√7, but distance can't be negative.

This is confusing.

Perhaps the problem requires a different approach.

Let me recall that for two surfaces to be tangent along a circle, the circle must lie on both surfaces, and their normals must be colinear at every point on the circle.

For the torus and sphere, this implies that the circle lies in a plane where the distance from the sphere's center to the plane is such that the sphere's radius in the plane is equal to the torus's radius in that plane.

The sphere's radius in the plane is sqrt(11² - d²), where d is the distance from the sphere's center to the plane.

The torus's radius in the plane depends on its orientation. If the torus's axis is perpendicular to the plane, then its radius in the plane is its minor radius, 3.

Therefore, setting sqrt(121 - d²)=3 ⇒ d= sqrt(112).

The distance from the sphere's center to the plane is sqrt(112).

The torus's generating circle must be positioned such that its center is at distance sqrt(112) ± something.

Alternatively, the distance from the torus's generating circle center to the plane must be such that the torus's cross-section has radius 3.

If the torus's axis is perpendicular to the plane, then the generating circle's center is at distance sqrt( (major radius)^2 + (distance to plane)^2 ).

Wait, this is getting too convoluted.

Given the time I've spent and the lack of progress, perhaps I should look for an alternative method or recall that the difference is (sqrt(11^2 - (6+3)^2) - sqrt(11^2 - (6-3)^2)).

Wait, that would be sqrt(121 -81)=sqrt(40)=2*sqrt(10) and sqrt(121-9)=sqrt(112)=4*sqrt(7). The difference would be 4√7 - 2√10, which is irrational.

Alternatively, perhaps the radii are 9 and 3, giving a difference of 6, but the problem states a fraction.

Alternatively, the problem might involve the radii of the circles on the torus where it is tangent. The major radius is 6, minor 3. The radii of the circles of tangency would be R ± r, so 9 and 3, but scaled by the sphere's geometry.

Alternatively, the radii are 9 and 3 multiplied by some factor based on the distance.

Wait, let me think of the torus as a surface where any point on the circle of tangency must satisfy both equations.

Let me assume that the circle of tangency lies in the plane z=0.

Then, the sphere's equation gives x² + y² =121,

the torus's equation gives (sqrt( (x-D)^2 + y^2 ) -6)^2 =9.

Solving these:

sqrt( (x-D)^2 + y^2 ) =6 ±3,

so,

(x-D)^2 + y^2 =81 or 9.

Setting equal to the sphere's equation:

For the first case (81):

x² -2Dx + D² + y² =81,

but x² + y²=121,

so,

121 -2Dx + D² =81,

=> -2Dx = -40 -D²,

=> x=(D² +40)/(2D).

This must hold for all x on the circle, which is only possible if D=0, which isn't valid.

Similarly for the second case (9):

x=(D² +112)/(2D).

Again, only possible if D=0.

Thus, no solution unless it's a single point.

This suggests that the circle of tangency cannot lie in the z=0 plane unless it's a point.

Therefore, the circle must lie in another plane.

Let me consider a general plane perpendicular to the line connecting the centers (the x-axis). Let the plane be at x = p.

In this plane, the sphere's cross-section is a circle of radius sqrt(121 - p²).

The torus's cross-section is given by:

( sqrt( (p - D)^2 + y^2 ) -6 )^2 + z^2 =9.

But in the plane x=p, z can vary, so the cross-section is more complex.

However, for the cross-section to be a circle, the equation must reduce to a circle.

Assume that in this plane, the torus's cross-section is a circle.

Let me set z=0 for simplicity (the widest part).

Then,

( sqrt( (p - D)^2 + y^2 ) -6 )^2 =9,

so,

sqrt( (p - D)^2 + y^2 ) =6 ±3,

thus,

(p - D)^2 + y^2 =81 or 9.

These are circles of radius 9 or 3 centered at (p - D,0).

To coincide with the sphere's circle x=p, y² + z²=121 -p²,

the radii must match:

For the radius 9 circle: 9 = sqrt(121 - p²),

so,

81 =121 -p² ⇒ p²=40 ⇒ p=±2√10.

Similarly for radius 3: 3= sqrt(121 -p²) ⇒ p²=112 ⇒ p=±4√7.

Thus, the possible distances p are ±2√10 and ±4√7.

The distance between the centers D must satisfy:

For the radius 9 case:

The generating circle's center is at D, so the distance from D to the plane x=p is |D - p|.

The radius of the torus's cross-section is 9, which is the major radius plus minor radius.

Thus, the distance between centers D and the plane's position p must satisfy:

The generating circle's center is at D, and the cross-section at x=p has radius 9, so:

sqrt( (p - D)^2 +0 ) +3 =9 ⇒ |p - D|=6.

Thus,

D = p ±6.

Since p=2√10≈6.324,

D=2√10 ±6.

Similarly for p= -2√10, but distance can't be negative.

Taking p=2√10,

D=2√10 +6 or D=2√10 -6.

Similarly for the radius 3 case:

The cross-section radius is 3, so:

sqrt( (p - D)^2 ) -3 =0 ⇒ |p - D|=3,

Thus,

D= p ±3.

With p=4√7≈10.583,

D=4√7 ±3.

Now, the distance between the sphere's center and the generating circle's center is D.

For the case of radius 9 (p=2√10):

If D=2√10 +6,

then the distance between centers is D=2√10 +6,

and the radius of the circle of tangency is 9.

For the case of radius 3 (p=4√7):

If D=4√7 -3,

then the distance between centers is D=4√7 -3,

and the radius is 3.

Thus, the two radii are 9 and 3, but scaled by the sphere's geometry?

Wait, no. The radii of the circles of tangency are the sphere's cross-sectional radii at those planes:

For p=2√10, the radius is 9,

for p=4√7, the radius is 3.

Thus, the radii r_i and r_o are 9 and 3, so the difference is 6, which is 6/1, so m+n=7.

But the problem states "the difference r_i - r_o can be written as m/n where m and n are coprime". If it's 6, then m=6, n=1, so 7. But maybe the radii are reversed.

Wait, when the torus is resting on the outside of the sphere, perhaps the radius is smaller (3), and when resting on the inside, it's larger (9). Thus, r_i (inside) is 9, r_o (outside) is 3, so difference is 6.

But the problem might have the difference as 9 -3=6.

However, the problem might consider the radii as the sphere's cross-sectional radii, which are 9 and 3, giving difference 6.

Alternatively, perhaps I made a mistake in assuming the radii are 9 and 3. The problem says "along a circle with radius r_i", which is the radius of the circle of tangency, which in the sphere's frame is 9 and 3.

Therefore, the difference is 9 -3=6, so m+n=6+1=7.

But I need to confirm.

Wait, let me check the problem again:

"When T rests on the outside of S, it is externally tangent to S along a circle with radius r_i, and when T rests on the outside of S, it is externally tangent to S along a circle with radius r_o."

Wait, the user might have a typo, but assuming it's "inside" for one of them, then the difference would be 6.

Alternatively, maybe the radii are sqrt(112) and sqrt(40), which are 4√7 and 2√10. Their difference is 4√7 -2√10. To write this as m/n, but it's irrational, so that can't be.

Alternatively, perhaps the problem considers the radii of the generating circles, which are 9 and 3, so difference 6.

Given the problem is from a competition, likely the answer is 6, so m+n=7.

But I need to verify once more.

Wait, in the case where the generating circle is at D=2√10 +6, the circle of tangency has radius 9 on the sphere.

Similarly, when D=4√7 -3, the radius is 3.

Thus, the difference is 9-3=6.

Therefore, the answer is 6/1, so m+n=7.

However, I recall that in the problem statement, the torus has major radius 6 and minor radius 3. When it's externally tangent on the outside, the radius would be smaller (3), and on the inside, larger (9). Thus, r_i (inside) is 9, r_o (outside) is 3, so difference is 6.

Therefore, the answer is 6, so m+n=7.

But I need to check if the problem specifies the order. It says "difference r_i - r_o". If r_i is the inside radius (9) and r_o is the outside (3), then 9-3=6.

Yes, that makes sense.

Therefore, the answer is 6, so m+n=6+1=7.

But I need to confirm once more.

Alternatively, perhaps the radii are calculated differently.

Wait, when the generating circle is at D=2√10 +6, the distance from the sphere's center to the plane is p=2√10, so the radius of the circle is sqrt(11² - p²)=sqrt(121-40)=sqrt(81)=9.

Similarly, for the other case, sqrt(121-112)=sqrt(9)=3.

Thus, the radii are indeed 9 and 3.

Therefore, the difference is 6, so the answer is 7.

But I need to check if the problem might have a different answer.

Wait, the problem says "when T rests on the outside of S", which might correspond to the smaller radius (3), and "rests on the inside" gives 9. Thus, difference 9-3=6.

Yes, so the answer is 6, so m+n=7.

However, I recall that in some problems, the radius of the circle of tangency on the torus is its minor radius, but here the problem specifies the radius along the sphere's surface.

Therefore, I think the correct answer is 6, so the final answer is 7.

**Final Answer**
\boxed{7}
</think>

To solve the problem, we need to determine the radii of the circles where the torus \( T \) is externally tangent to the sphere \( S \) in two different configurations (resting on the outside and inside) and find the difference between these radii.

1. **Torus and Sphere Setup**:
   - The torus \( T \) has a major radius \( R = 6 \) and a minor radius \( r = 3 \).
   - The sphere \( S \) has a radius of 11.

2. **Key Equations**:
   - The equation of the torus centered at \( (D, 0, 0) \) is:
     \[
     \left( \sqrt{(x - D)^2 + y^2} - 6 \right)^2 + z^2 = 9
     \]
   - The equation of the sphere centered at the origin is:
     \[
     x^2 + y^2 + z^2 = 121
     \]

3. **Tangency Conditions**:
   - For the torus and sphere to be tangent along a circle, the distance between their centers \( D \) must be such that the circles formed by their intersection are tangent.
   - The possible distances \( D \) between the centers are found by considering the external and internal tangency of the generating circle of the torus (radius 3) with the sphere (radius 11):
     - External tangency: \( D = 11 + 3 = 14 \)
     - Internal tangency: \( D = 11 - 3 = 8 \)

4. **Radii of Tangency Circles**:
   - For \( D = 8 \) (internal tangency), the radius of the circle of tangency on the sphere is:
     \[
     \sqrt{11^2 - 8^2} = \sqrt{121 - 64} = \sqrt{57}
     \]
   - For \( D = 14 \) (external tangency), the radius of the circle of tangency on the sphere is:
     \[
     \sqrt{11^2 - 14^2} \quad \text{(imaginary, thus invalid)}
     \]
   - Correcting the approach, we consider the valid radii from the cross-sectional analysis:
     - When the generating circle is at \( D = 2\sqrt{10} + 6 \), the radius is 9.
     - When the generating circle is at \( D = 4\sqrt{7} - 3 \), the radius is 3.

5. **Final Calculation**:
   - The valid radii of the circles of tangency are 9 and 3.
   - The difference \( r_i - r_o = 9 - 3 = 6 \).
   - Expressing this difference as \( \frac{m}{n} \), we get \( \frac{6}{1} \), so \( m = 6 \) and \( n = 1 \).

Thus, the sum \( m + n = 6 + 1 = 7 \).

\boxed{7}