Prove that the determinant represents a straight line. Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$, then show that the equation

represents a straight line.
Expansion of determinant is tedious. I can see that matrix is symmetric but not able to use this fact. Could someone help me with this?

\begin{eqnarray*}
 Det\left[
\begin{array}{ccc}
ax-by-c&bx+ay&cx+a \\
bx+ay&-ax+by-c&cy+b\\
cx+a&cy+b&-ax-by+c \\
\end{array}
\right]=(x^2+y^2+1)(a^2+b^2+c^2)(ax+by+c).
\end{eqnarray*}
The first two factors are positive so $\color{red}{ax+by+c=0}$.
