How Do You Determine the Equation and Tangency Point of a Sphere on the X-Axis?

[physstudent1]

Homework Statement

Find the equation of a sphere of radius 9 which is tangent to the plane x-2y+2z=4 and whose center lies on the x axis.
b. Determine the point where the sphere is tangent to the plane.

Homework Equations

The Attempt at a Solution

My TA told me it might be easier to solver for b first, so I think that the the normal vector of the plane is going to be the same as one of the radii vectors of the sphere, but I really don't know what to do from here could someone point me in the right direction please.

 

[mighty2000]

Hello,

Firstly, let's define some variables to make the problem easier to understand. Let the center of the sphere be denoted as (a, 0, 0) where a is the distance between the center and the origin on the x-axis. Let the point of tangency between the sphere and the plane be denoted as (x, y, z).

Now, we can use the equation of the sphere to find the relationship between these variables:
(x-a)^2 + y^2 + z^2 = 9^2

Next, we can use the equation of the plane to find the relationship between (x, y, z):
x - 2y + 2z = 4

Since the sphere is tangent to the plane, we know that the distance between the center of the sphere and the plane is equal to the radius of the sphere (9). Therefore, we can use the distance formula to set up an equation:

Distance between (a, 0, 0) and (x, y, z) = 9
sqrt((x-a)^2 + y^2 + z^2) = 9

Now, we can substitute the equation of the plane into this equation to solve for x:
sqrt((4+2y-2z-a)^2 + y^2 + z^2) = 9

Squaring both sides and simplifying, we get:
(x - a)^2 + 2y^2 + 2z^2 - 2xy + 4yz - 4xz = 0

Since we know that the center of the sphere lies on the x-axis, we can set a = 0. This simplifies the equation to:
x^2 + 2y^2 + 2z^2 - 2xy + 4yz = 0

Now, we can use the equation of the plane to eliminate one of the variables. Solving for z in terms of x and y, we get:
z = 2 - (x/2) + (y/2)

Substituting this into the equation above, we get:
x^2 + 2y^2 + (4 - x + y)^2 - 2x(4 - x + y) + 4y(4 - x + y) = 0

Simplifying