Perpendicularity and Coincidence Conditions

The angle formula derived in the previous section, $\tan\theta=\left|\dfrac{2\sqrt{h^2-ab}}{a+b}\right|$, has two important special cases that are asked about far more often than the general angle itself: when the lines are perpendicular ($\theta=90^\circ$) and when they coincide ($\theta=0^\circ$, really the same line twice).

Perpendicularity. From $\tan\theta = \left|\dfrac{2\sqrt{h^2-ab}}{a+b}\right|$, the angle is $90^\circ$ exactly when this ratio is undefined, i.e. when its denominator vanishes while the numerator stays finite and non-zero:

This can also be seen directly from the slopes: $m_1m_2 = a/b$, and two lines are perpendicular exactly when $m_1m_2=-1$, i.e. $a/b=-1$, i.e. $a+b=0$. Notice that this condition depends only on the coefficients of $x^2$ and $y^2$ — the coefficient $h$ of $xy$ plays no role at all in whether the lines are perpendicular, only in what the actual lines and their angle are otherwise. So, for example, the lines represented by $\lambda x^2+8xy+2y^2=0$ are perpendicular for exactly one value of $\lambda$, found purely by solving $\lambda+2=0$, i.e. $\lambda=-2$, regardless of the middle coefficient $8$.

Coincidence. Two lines through the origin coincide (i.e. the "pair" is really one line, repeated) exactly when the discriminant of the underlying quadratic is zero, which — as established in the real-and-distinct-lines section — is precisely

Geometrically, when $h^2=ab$ the homogeneous expression becomes a perfect square: $ax^2+2hxy+by^2 = b\left(y - my\right)^2$ for the repeated slope $m=-h/b$, so the "combined equation" is really $b(y-mx)^2=0$, a single line counted with multiplicity two. As a worked check, find $\lambda$ so that $x^2+\lambda xy+9y^2=0$ represents a coincident pair: here $a=1$, $b=9$, $h=\lambda/2$, so $h^2=ab$ becomes $\lambda^2/4 = 9$, giving $\lambda^2=36$, i.e. $\lambda=\pm6$. …