"Conic sections" that are really just two straight lines

Question

My teacher was teaching co-ordinate geometry and today he said that the following equation will always represent a conic section:$$ax^2+by^2+2hxy+2gx+2fy+c=0$$ Then he said that if the determinant of the following matrix is zero then the equation can be split into two linear factors, and thus will represent a couple of straight lines.$$ \begin{bmatrix}a & h & g\\h & b & f\\ g & f & c\end{bmatrix}$$ What is the proof of this theorem?

Answer 1

When a conic consists of two lines, moreover called a degenerate conic, it can be written as follows:

$$ (ax+by+c)(dx+ey+f)=0 $$

Its determinant is (MathGem's hint 2):

$$ \frac{1}{8}\begin{vmatrix} 2ad & ae+bd & af+cd \\ ae+bd & 2be & bf+ce \\ af+cd & bf+ce & 2cf\end{vmatrix} $$

which is $0$.

Answer 2

Hint:

1. Ask yourself when a conic section becomes degenerate?

2. Calculate the determinant of that matrix.

Can you see a relation between 1 and 2?