Find the points on the hyperboloid 9x^2 -45y^2+5z^2 = 45 where the tangent plane is parallel to x+5y-2z = 7.
The Attempt at a Solution
Ok so the tangent plane is parallel to the x+5y-2z=7 when their normal vectors are parallel. So that means that the normal vector for the given plane (1,5,-2) should also be a normal vector for the tangent plane to the hyperboloid.
So since the equation of the plane can be written as
(1,5,-2) . (x-x0,y-y0,z-z0) = x - x0 + 5y -5y0 -2z + 2z0 where -x0 - 5y0 +2z0 = -7
and the equation of the tangent plane to the hyperboloid can be written as the gradient at (x0,y0,z0) . (x-x0,y-y0,z-z0)
the gradient is (18x0, -90y0, 10z0) at (x0,y0,z0). and it has to be equal to the normal vector (1,5,-2).
So 18x0 = 1, -90y0 = 5, 10z0 = -2? and then solve for x0,y0,z0 and that is the point I'm looking for?
Just wanted to know if I'm doing this right.