287. How many solutions does the system of equations generally have

$$
\begin{gathered}
a x^{2}+b x y+c y^{2}=d \\
a_{1} x^{2}+b_{1} x y+c_{1} y^{2}=d_{1} ?
\end{gathered}
$$

In particular, how many solutions does the system of question 280 have?
287. Generally four. Excluding $d$ and $d_{1}$, we obtain a homogeneous equation of the second degree, which, generally, will determine two values of the ratio $\frac{x}{y}$ (see question 284) $\frac{x}{y}=k_{1}$ and $\frac{x}{y}=k_{2}$, from which $x=k_{1} y, x=k_{2} y$. Substituting each of these expressions sequentially into one of the given equations, we will obtain a quadratic equation each time, which will determine two values for $y$, and consequently for $x$. In total, generally, four solutions will be obtained.