Question

How do you find an equation of the sphere that passes through the point (9, 4,-6) and has a center (6,7,2)?

Answer 1

Find the equation for the equivalent sphere centered at (0,0,0) and then shift the co-ordinates.

`(9,4,-6)` relative to a center `(6,7,2)`
is equivalent to `(3,-3,-8)` relative to `(0,0,0)`

The equation of a sphere through `(3,-3,-8)` with center `(0,0,0)` is

`X^2 + Y^2 + Z^2 = (3)^2 + (-3)^2 + (-8)^2`
or
`X^2 + Y^2 + Z^2 = 82`

For `(x,y,z) = (6,7,2)` to be equivalent to `(X,Y,Z) = (0,0,0)`
`X = x-6`
`Y = y-7`
`Z = z-2`

So the desired equations is:
`(x-6)^2 + (y-7)^2 + (z-2)^2 = 82`