Suppose normal lines are drawn at all the points on the surface $z = ax^2 + by^2$, where $a$ and $b$ are some positive constants, that are at a given height $h$ above the $xy$-plane. Find an equation of the curve (in terms of $x$ and $y$) consisting of all the intersection points of these normal lines and the $xy$-plane.
So the above is the question in one of my tutorials. Honestly I don't have a clue how to begin, judging of the description, I'm guessing this curve might be the projection of the surface on the $xy$ plane, but not sure how to proceed from there.
Any help and insights is deeply appreciated.
$X=(x,y,ax^2+by^2)$ so that $$ X_x=(1,0,2ax),\ X_y=(0,1,2by) $$
Hence let $$ N=X_x\times X_y=(-2ax,-2by,1) $$
If $ax^2+by^2=h$, $$ X-hN=((1+2a)x,(1+2b)y,0)\ (:=(u,v,0)) $$
So $$ h=ax^2+by^2=a (\frac{u}{1+2a})^2 + b(\frac{v}{1+2b})^2 $$