Finding the Center and Radius of a Sphere with Parallel Tangent Planes

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The sphere has two parallel tangent planes with equations x+2y-2z=37 and x+2y-2z=-11, with a point of tangency at (-7,5,7) on the lower plane. The distance from this point to the upper plane is calculated to be 16, indicating that the sphere's radius is 8. To find the center of the sphere, one must determine the normal direction of the planes and move 8 units from the point of tangency towards the upper plane. This approach ensures that the center is correctly positioned between the two tangent planes. The discussion emphasizes the geometric relationship between the sphere and the tangent planes.
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A sphere has two parallel tangent planes with equations x+2y-2z=37 and x+2y-2z=-11. One of the points of tangency is (-7,5,7). Find the center and radius of the sphere.


I'm not really sure how to do this. I know that the point (-7,5,7) lies on the x+2y-2z=-11 plane. The distance from the point to the plane using the distance formula is 16. So now I know that the sphere's diameter is 16, making the radius 8.

Can someone explain how to find the center of the sphere?

Thanks!
 
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That's good so far. Now find the normal direction of the planes. That means the center of the sphere lies 8 units in the direction of the normal from (-7,5,7) towards the other plane.
 

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