# Applying the Implicit Function Theorem to find derivatives

I´m solving the problem:

The intersection of $2x^2+3y^2-z^2=25$ and $x^2+y^2=z^2$ contains a curve $C$ that passes through the point $(7^{1/2},3,4)$. Find $\frac { \partial x}{\partial z}$ and $\frac { \partial y}{\partial z}$.

I was suggested to use the Implicit Function Theorem (IFT); so I need to prove that $C(7^{1/2},3,4)=0$, which is easy since the point is in the curve, and that the Jacobian matrix of $C$ is not zero, which I'm not sure how to compute for this case. Also, once I show that I can actually use the IFT, how should I proceed from there?

by an algebraic manipulation you will get $2z^2-25=x^2$ and $z^2+y^2=25$ so you can find your derivative.

• Should I say that $C_x$ and $C_y$ are both different from zero to justify the fact that I can compute this derivatives? – Weierstraß Ramirez Oct 4 '14 at 18:55
• Can I just derivate each of those two expressions implicitly? – Weierstraß Ramirez Oct 4 '14 at 20:19