You will often be asked to show that a plane is tangent to a spherical surface and to find the tangent point. A plane is tangent to a spherical surface if the distance from the center of the sphere to the plane is equal to the radius of the sphere.
Say you have the sphere
and the plane
From the equation of the sphere, you can see that the radius is and the center is at . To show that the plane is tangent to the surface of the sphere, you use the formula for the distance between a point and a plane and check whether the answer is equal to the radius of the sphere, which we know is :
To find the tangent point where the surface of the sphere and the plane touches, you can do like in Example 1. You can also use the formula above for the tangent point of a spherical surface and a plane.
Note! Take a note of that in this case, the normal vector can send you in the wrong direction. Then you will end up at the opposite side of the sphere. For that reason, check whether your answer lies in the tangent plane. If it doesn’t, simply switch with .
In this case, the radius of the sphere and the distance between the center of the sphere and the plane are the same, giving you that
Put this into the equation of the tangent plane to check whether you have gone in the right direction. That gives you
You can see that this does not satisfy the equation of the plane, which means the normal vector has sent you in the wrong direction. If you change the sign in front of the normal vector, you get
Put this into the equation of the plane, and you get