Question 62
Let $k$ be a real parameter. Determine the number of real solutions to the system
$$
\begin{align*}
x^2 + kxy + y^2 &= z, \\
y^2 + kyz + z^2 &= x, \\
z^2 + kzx + x^2 &= y,
\end{align*}
$$
in terms of $k$.
$$
\begin{align*}
x^2 + kxy + y^2 &= z, \\
y^2 + kyz + z^2 &= x, \\
z^2 + kzx + x^2 &= y,
\end{align*}
$$
in terms of $k$.