Solving Hyperboloid Problem: Finding Points of Parallel Normal Line

[bodensee9]

Homework Statement

Hello, can someone tell me where I went wrong? So, I am supposed to find the points on the hyperboloid x^2-y^2+2z^2 = 1 where the normal line is parallel to the line that joins the points (3,-1,0) and (5,3,6).

Homework Equations

I think I'm supposed to find the gradient vector of the hyperboloid, because that has the direction of the normal line.

The Attempt at a Solution

So, if I'm right, the gradient of the hyperboloid is <2x, -2y, 2z>, and that is supposed to have the same direction as the vector <2, 4, 6>. So, wouldn't I just be looking for values of x, y, and z that is in multiples of <2, 4, 6>? That seems wrong to me. Thanks!

 

[AKG]

Wouldn't the gradient be <2x, -2y, 4z>? Then you'd need to find (x,y,z) such that <2x, -2y, 4z> is a scalar multiple of <2, 4, 6> AND (x,y,z) must lie on the given hyperboloid, i.e. it must satisfy x² - y² + 2z² = 1.