Condition for Straight Lines in Plane to be Perpendicular/General Equation/Corollary
Let $L$ be described by the general equation for the straight line:
- $l x + m y + n = 0$
- $m x - l y = k$
From the general equation for the straight line, $L$ can be expressed as:
- $y = -\dfrac l m x + \dfrac n m$
Hence the slope of $L$ is $-\dfrac l m$.
Let $L'$ be perpendicular to $L$.
From Condition for Straight Lines in Plane to be Perpendicular, the slope of $L'$ is $\dfrac m l$.
Hence $L'$ has the equation:
- $y = \dfrac m l x + r$
for some $r \in \R$.
Hence by multiplying by $l$ and rearranging:
- $m x - l y = -l r$
The result follows by replacing the arbitrary $-l r$ with the equally arbitrary $k$.