Find the tangent point of a sphere and a ellipsoid

[whtseng]

Here is a problem in my calculus test which I've spent a lot of time but still can't make it:

Find the point that the sphere x²+y²+z²-8x-8y-6z+24=0 is tangent to the ellipsoid x²+3y²+2z²=9.



And the following is how I've tried:

(2x-8) i + (2y-8) j + (2z-6) k = λ[(2x) i + (6y) j + (4z) k] (i, j, k are vectors)

then, 2x-8 = λ2x, 2y-8 = λ6y, 2z-6 = λ4z,
y = (2x)/(6-x), z = (3x)/(8-x)


BUT, when I replace y and z in the sphere function and ellipsoid function, I just can't solve the x.
Can you please tell me which part I'm doing wrong, or I'm using a wrong method?

I really appreciate your help!

 

[galactus]

Let \(\displaystyle \lambda\) = -1 in your equations and solve for x, y, z.

 

[whtseng]

THANK YOU!! I got it.