# What is the equation of the line that passes through (6,-4) and is perpendicular to the line that passes through the following points: (-2,12),(5,-6) ?

Jan 10, 2018

See a solution process below:

#### Explanation:

First, we need to determine the slope of the line passing through $\left(- 2 , 12\right)$ and $\left(5 , - 6\right)$. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{- 6} - \textcolor{b l u e}{12}}{\textcolor{red}{5} - \textcolor{b l u e}{- 2}} = \frac{\textcolor{red}{- 6} - \textcolor{b l u e}{12}}{\textcolor{red}{5} + \textcolor{b l u e}{2}} = - \frac{18}{7}$

Let's call the slope of a perpendicular line: ${m}_{p}$

Then the rule for find the slope of a perpendicular line is:

${m}_{p} = - \frac{1}{m}$

Substituting gives: ${m}_{p} = \frac{- 1}{- \frac{18}{7}} = \frac{7}{18}$

We can now use the point slope formula to find an equation of the line for the point given in the problem and the slope we calculated. The point-slope form of a linear equation is: $\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

Where $\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ is a point on the line and $\textcolor{red}{m}$ is the slope.

Substituting again gives:

$\left(y - \textcolor{b l u e}{- 4}\right) = \textcolor{red}{\frac{7}{18}} \left(x - \textcolor{b l u e}{6}\right)$

Or

$\left(y + \textcolor{b l u e}{4}\right) = \textcolor{red}{\frac{7}{18}} \left(x - \textcolor{b l u e}{6}\right)$