Question

What is the equation of the line that passes through `(2,1)` and is perpendicular to the line that passes through the following points: `(8,-5),(3,9)`?

Answer 1

Let `A=(2,1), B=(8,-5), C=(3,9)`

First of all we have to find out the slope of the line that passes through `B` and `C`.
After knowing the slope of the line passing through `B` and `C` we can find the slope of the line that is perpendicular to `BC`.

Slope of `BC=(9-(-5))/(3-8)=(9+5)/-5=12/-5=-12/5=m`

Let the slope of the line perpendicular to `BC` be `m'`.
Then `m.m'=-1`

`implies m'=-1/m=-1/(-12/5)=5/12`

`implies m'=5/12`

Now, we know two things about the required line
(i) Slope of that line `m'=5/12`

(ii) The point through which the line passes `(2,1)`.

These two things are enough to find the equation.

Using point-slope form

`y-y_1=m'(x-x_1)`

`implies y-1=5/12(x-2)`

`implies 12y-12=5x-10`
`implies 5x-12y+2=0`

This is the required equation.