Let $\mathbb{F}_{q}$ be the finite field of order $q$ where $q$ is the power of an odd prime. Let $u$ be a fixed non-square in $\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q}$ such that $\lambda\neq\pm 1$. Consider the two quadrics defined by $f_{1}$ and $f_{2}$ as follows:
$$f_{1}(x,\mu):=ux^2-\mu^2-2\lambda^2\mu-2\lambda^2+1,$$ $$f_{2}(y,\mu):=uy^2-\mu^2+2\lambda^2\mu-2\lambda^2+1.$$
I would like to show that there exists $\mu, x, y\in\mathbb{F}_{q}$ such that $(x,\mu)$ and $(y,\mu)$ are solutions to $f_{1}$ and $f_{2}$ respectively. So I would like to find the intersection of the two quadrics and show that it contains lots of points.
How would I go about finding the equation which defines the intersection of the two quadrics? Moreover, how do I determine whether the intersection will be a curve, and more specifically an elliptic curve?
I have tried combining the equations for $f_{1}$ and $f_{2}$, but the resulting equation is not in the form $t^{2}=\textrm{cubic}$ or $t^{2}=\textrm{quadratic}$, so I am not sure how to determine whether the curve will be elliptic or not.
I would really appreciate any hints. Thank you!
The standard method is the following: parametrize one quadric and plug the result into the second. Weil in his book on number theory from Hammurabi to Legendre devotes a couple of pages to this problem.