# How do you write an equation in point slope form given (p,q), (-p, 2q)?

Jan 16, 2017

$\left(y - \textcolor{red}{q}\right) = \textcolor{b l u e}{- \frac{q}{2 p}} \left(x - \textcolor{red}{p}\right)$

or

$\left(y - \textcolor{red}{2 q}\right) = \textcolor{b l u e}{- \frac{q}{2 p}} \left(x + \textcolor{red}{p}\right)$

#### Explanation:

First, we need to determine the slope from the two points given in the problem.

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 problem gives:

$m = \frac{\textcolor{red}{2 q} - \textcolor{b l u e}{q}}{\textcolor{red}{- p} - \textcolor{b l u e}{p}}$

$m = \frac{q}{-} 2 p = - \frac{q}{2 p}$

Now we can use either point to build a equation in the point-slope form:

The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

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

Solution for point (p, q):

$\left(y - \textcolor{red}{q}\right) = \textcolor{b l u e}{- \frac{q}{2 p}} \left(x - \textcolor{red}{p}\right)$

Solution for point (-p, 2q):

$\left(y - \textcolor{red}{2 q}\right) = \textcolor{b l u e}{- \frac{q}{2 p}} \left(x - \textcolor{red}{- p}\right)$

$\left(y - \textcolor{red}{2 q}\right) = \textcolor{b l u e}{- \frac{q}{2 p}} \left(x + \textcolor{red}{p}\right)$