Homogeneous Quadratic Equation representing Coincident Straight Lines

Theorem

Let $E$ be a homogeneous quadratic equation in two variables:

$E: \quad a x^2 + 2 h x y + b y^2$

Let $h^2 - a b = 0$.

Then

Then $E$ represents $2$ straight lines in the Cartesian plane which completely coincide:

Proof

From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $E$ represents $2$ straight lines in the Cartesian plane:

$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$

But when $h^2 - a b = 0$, we get:

$y = \dfrac h b x$

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $15$. The homogeneous equation of second degree