Explicit bivariate resultant example

Question

I have two polynomials modulo a prime $p$ with a common root $(x_0,y_0)$ with $|x_0|,|y_0|<\sqrt{p}$ and there are no other roots in $\mathbb F_p$ or no other roots of this size in $\mathbb F_p$ (two scenarios).

1. $a_1x^2+b_1xy+c_1y^2=d_1\bmod p$

2. $a_2x^2+b_2xy+c_2y^2=d_2\bmod p$

How to use the tools of resultant theory to get it down to a single variable polynomial which gets the root $x_0$ or $y_0$?