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

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In summary, the given equations represent two parallel tangent planes of a sphere. One of the points of tangency is (-7,5,7) and the distance from this point to one of the planes is 16. This means the sphere's diameter is 16 and the radius is 8. To find the center of the sphere, we need to find the normal direction of the planes. 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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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.
 

1. What is a sphere?

A sphere is a three-dimensional shape with all points on its surface equidistant from its center. It is a type of geometric object that is commonly referred to as a "perfect" sphere, as it is perfectly symmetrical and has no edges or corners.

2. How do you find the equation of a tangent plane to a sphere?

To find the equation of a tangent plane to a sphere, you need to know the coordinates of the center of the sphere and the radius of the sphere. Then, you can use the formula (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2, where (x0, y0, z0) are the coordinates of the center and r is the radius. This equation represents the equation of the sphere. To find the equation of the tangent plane, you can use the gradient vector of the sphere's equation and the coordinates of a point on the sphere's surface. This will give you the equation of the tangent plane in the form ax + by + cz = d.

3. How many tangent planes can be drawn to a sphere?

There can be an infinite number of tangent planes drawn to a sphere. This is because any plane that touches the sphere at exactly one point is considered a tangent plane. Since the surface of a sphere has no edges or corners, there is no limit to the number of tangent planes that can be drawn.

4. What is the relationship between a tangent plane and a normal vector to a sphere?

A tangent plane to a sphere is always perpendicular to the normal vector of the sphere at the point of tangency. This means that the normal vector of the sphere is always orthogonal (or perpendicular) to the tangent plane at the point of contact.

5. How are tangent planes used in real life applications?

Tangent planes are used in various applications, such as computer graphics, engineering, and physics. In computer graphics, tangent planes are used to create realistic 3D models of objects, including spheres. In engineering, they are used to calculate the direction of forces on curved surfaces, such as spheres. In physics, they are used to understand the behavior of objects in motion and the effects of forces on these objects.

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