Equations of Perpendicular Lines with the Same Y-Interception

# Equations of Perpendicular Lines with the Same Y-Interception

Let $ax + by + c = 0$ be the equation of a line where $a, b ≠ 0$. We are restricting $a$ and $b$ right now as if one of $a$ or $b$ is zero, we have either a horizontal line (if $a = 0$) or a vertical line (if $b = 0$). We will look at these cases later.

If we take $ax + by + c = 0$ and convert it to slope-intercept form, we get that $y = \frac{-a}{b}x - \frac{c}{b}$. To find a formula for a perpendicular line that has the same y-intercept, we note our line must $m = \frac{b}{a}$ and $b = -\frac{c}{b}$. Thus:

(1)
\begin{align} y = bx - \frac{c}{b} \\ ay = bx - \frac{ac}{b} \\ (ab)y = b^2x - ac \\ b^2x - (ab)y - ac = 0 \end{align}

Thus if $ax + by + c = 0$ is a line $L$, then $b^2x - (ab)y - ac = 0$ is a line that is perpendicular to $L$ and intersects $L$ at the $y$-axis.