Characteristic of Quadratic Equation that Represents Two Straight Lines

Theorem
Consider the quadratic equation in $2$ variables:


 * $(1): \quad a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

where $x$ and $y$ are independent variables.

Then $(1)$ represents $2$ straight lines its discriminant equals zero:


 * $a b c + 2 f g h - a f^2 - b g^2 - c h^2 = 0$

This can also be expressed in the form of a determinant:
 * $\begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix} = 0$

Proof
Suppose that $a \ne 0$.

We have:

In order that the second part is a perfect square in $y$, it is necessary that:

Conversely, if $(3)$ is true, then $(2)$ can be expressed in the form of a Difference of Two Squares:

Hence $(2)$ has $2$ factors, which can be seen to be the equations of straight lines.

Let $a = 0$ but $b \ne 0$.

Then:

In order that the second part is a perfect square in $x$, it is necessary that:

it is noted that $(4)$ is the same as $(3)$ but with $a = 0$.

Suppose $a = 0$ and $b = 0$ but $h \ne 0$.

Then:

and it is seen that in order for $(1)$ to be divisible into the $2$ required factors:


 * $2 \paren {h x + f} \paren {h y + g} = 0$

it is necessary for $c h = 2 f g$.

This is again the same as $(3)$ when you set $a = 0$ and $b = 0$.

If $a = 0$ and $b = 0$ and $h = 0$, then $(1)$ is not a quadratic equation.

All cases have been covered.

Finally we see that: