Product of equation of two lines Why does the product of equations of straight lines represents 2 straight lines?
Why does a 2nd degree equation is so linear?

In the language of conics,
\begin{align}
  0 &= (ax+by+c)(a'x+b'y+c') \\[10pt]
  \iff 0 &=
  \begin{pmatrix}
    x & y & 1
   \end{pmatrix}
   \begin{pmatrix}
     aa' & \frac{ab'+a'b}{2} & \frac{ac'+a'c}{2} \\
     \frac{ab'+a'b}{2} & bb' & \frac{bc'+b'c}{2} \\
     \frac{ac'+a'c}{2} & \frac{bc'+b'c}{2} & cc'
   \end{pmatrix}
   \begin{pmatrix}
    x \\ y \\ 1
   \end{pmatrix}
\end{align}
Note that
$$\det
\begin{pmatrix}
  aa' & \frac{ab'+a'b}{2} \\
  \frac{ab'+a'b}{2} & bb'
\end{pmatrix}
=-\frac{(ab'-a'b)^2}{4}<0$$
providing non-parallel and
$$\det
\begin{pmatrix}
  aa' & \frac{ab'+a'b}{2} & \frac{ac'+a'c}{2} \\
  \frac{ab'+a'b}{2} & bb' & \frac{bc'+b'c}{2} \\
  \frac{ac'+a'c}{2} & \frac{bc'+b'c}{2} & cc'
\end{pmatrix}
=0$$
implying a pair of straight lines.

It's a special case that the section plane cutting the radiant point of the cones.


See another answer here for your interest.
