# Perpendicular Lines Shortcut

You probably know how to solve this since solving perpendicular lines is very easy. Problem solving about this topic and its application was published few months ago in this page. But now let’s talk about speed and accuracy which is the prime purpose of a shortcut.

Derivation:

Given a line with equation  $ax+by+c=0$. The equation of the line passing through point $P(x_1,y_1)$  and perpendicular to it can be solved as follows.

Two lines are perpendicular if and only if the product of their slopes is equal to -1.

Solving for the slope of the given line

$ax+by+c=0$

$by=- ax-c$

$y=\displaystyle\frac{-ax-c}{b}$

$y=\displaystyle\frac{-ax}{b}-\displaystyle\frac{-c}{b}$

The slope of the line in the form of $y=mx+b$ is m. Hence, the slope of the line here is $\displaystyle\frac{-a}{b}$

Let $m_2$ be the slope of the perpendicular line.

Let $m_1$ be the slope of the given line.

Solving for the slope of the perpendicular line we have,

$m_1m_2=-1$

$\displaystyle\frac{-a}{b}m_2=-1$

$m_2=\displaystyle\frac{b}{a}$

Solving for the equation of perpendicular line we have

$y-y_1=m_2(x-x_1)$

$y-y_1=\displaystyle\frac{b}{a}(x-x_1)$

$a(y-y_1)=bx-bx_1$

$ay-ay_1=bx-bx_1$

$bx-ay=bx_1-ay_1$

This means that the equation of the line passing through $(x_1,y_1)$ and perpendicular to the line $ax_1+by_1+c=0$  is  $bx-ay=bx_1-ay_1$.

Knowing this shortcut, you don’t need to calculate that much to solve for it. You just need to create your own mnemonics to effectively remember this shortcut so that you can pull it out of your head anytime you need it.

Worked Problem 1:

Find the equation of the line through (1,2) and perpendicular to the line $2x+y=5$.

Solution:

Using the shortcut formula we have,

$bx-ay=bx_1-ay_1$

$a=2,b=1,x_1=1,y_1=2$

$(1)x-(2)y=(1)(1)-(2)(2)$

$x-2y=-3$

Or

$x-2y+3=0$

Worked Problem 2:

Find the equation of the line passing through  $(-2,1)$  and perpendicular to $2x-3y=4$.

Solution:

Using the derived shortcut we have,

$bx-ay=bx_1-ay_1$

$(-3)x-(2)y=(-3)(-2)-(2)(1)$

$-3x-2y=-8$

$3x+2y-8=0$