Angle Between a Pair of Lines and Special Conditions

For the pair of lines represented by $ax^2+2hxy+by^2=0$, the angle $\theta$ between them can be found directly from $a$, $h$, $b$ without factoring first:

Two special cases are worth remembering because they turn up often in problems:

• The two lines are perpendicular exactly when the coefficient of $x^2$ plus the coefficient of $y^2$ is zero: $a + b = 0$. (When $a+b=0$ the angle formula above is undefined because the lines are at $90°$ — the denominator vanishing is exactly the signal.)
• The two lines coincide exactly when $h^2 = ab$, as already noted in the previous section. …