# What is the equation of the line perpendicular to y=-7x  that passes through  (6,-1) ?

Mar 18, 2017

$y = \frac{1}{7} x - \frac{13}{7}$

#### Explanation:

In general an equation of the form
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} x + \textcolor{b l u e}{b}$
has a slope of $\textcolor{g r e e n}{m}$

$y = \textcolor{g r e e n}{- 7} x$ is equivalent to $y = \textcolor{g r e e n}{- 7} x + \textcolor{b l u e}{0}$
and thus has a slope of color(green)(""(-7))

If a line has a slope of $\textcolor{g r e e n}{m}$ then all lines perpendicular to it have a slope of color(magenta)(""(-1/m))

Therefore any line perpendicular to $y = \textcolor{g r e e n}{- 7} x$
has a slope of $\textcolor{m a \ge n t a}{\frac{1}{7}}$

If such a perpendicular line passes through the point $\left(\textcolor{red}{x} , \textcolor{b r o w n}{y}\right) = \left(\textcolor{red}{6} , \textcolor{b r o w n}{- 1}\right)$
we can use the slope-point formula:
$\textcolor{w h i t e}{\text{XXX}} \frac{y - \left(\textcolor{b r o w n}{- 1}\right)}{x - \textcolor{red}{6}} = \textcolor{m a \ge n t a}{\frac{1}{7}}$

Simplifying,
$\textcolor{w h i t e}{\text{XXX}} 7 y + 7 = x - 6$
or
$\textcolor{w h i t e}{\text{XXX")y=1/7x-13/7color(white)("XX}}$(in slope-intercept form)

Mar 18, 2017

$x - 7 y - 13 = 0.$

#### Explanation:

Slope of the line $L : y = - 7 x$ is $- 7.$

Knowing that, the Product of Slopes of mutually $\bot$ lines is

$- 1$, the slope of the reqd. $\bot$ line $\left(- \frac{1}{-} 7\right) = \frac{1}{7.}$

Also, the reqd. line passes thro. the pt. $\left(6 , - 1.\right)$

Therefore, by the Slope-Point Form, the eqn. of reqd. line is,

$y - \left(- 1\right) = \frac{1}{7} \left(x - 6\right) , i . e . , 7 y + 7 = x - 6.$

$\therefore x - 7 y - 13 = 0.$

Enjoy Maths.!