Question

What is the slope of any line perpendicular to the line passing through `(3,12)` and `(-5,17)`?

Answer 1

Of any line ?

`A = (3,12)` `B = (-5,17)`

`vec(AB) = (-5-3,17-12) = (-8,5)`

The equation of the line directed by this vector is `P = 5x+8y=0`

Now imagine all the couple which are solutions to this equation

`lambda = (x_0,x_1,...x_n;y_0,y_1,...y_n)`

Note that `A,B in lambda`

Now imagine an arbitrary coordinate `M(x,y)` It can be anything

`vec(lambdaM)` is perpendicular to `P` if and only if it is perpendicular to `vec(AB)` and it is perpendicular to `vec(AB)` if and only if `vec(lambdaM)*vec(AB) = 0`

`-8(x-x_0)+5(y-y_0) = 0` if you take the point `A` you have

`-8(x-3)+5(y-12) = 0`

if you take the point `B` you have :

`-8(x+5)+5(y-17) = 0`

...