Question

What is the equation of the line that is perpendicular to the line passing through `(-5,-6)` and `(4,-10)` at midpoint of the two points?

Answer 1

Equation of the line `18x-8y=55`

Explanation:

From the given two points `(-5, -6)` and `(4, -10)`, we need to obtain first the negative reciprocal of the slope m and the midpoint of the points.

Let start with the midpoint `(x_m, y_m)`

`x_m=(x_1+x_2)/2=(-5+4)/2=-1/2`
`y_m=(y_1+y_2)/2=(-6+(-10))/2=-8`

midpoint `(x_m, y_m)=(-1/2, -8)`

Negative reciprocal of the slope `m_p=-1/m`

`m_p=-1/m=(-1)/((-10--6)/(4--5))=(-1)/(-4/9)=9/4`

The equation of the line

`y-y_m=m_p(x-x_m)`

`y--8=9/4(x--1/2)`
`y+8=9/4(x+1/2)`
`4y+32=9x+9/2`

`8y+64=18x+9`

`18x-8y=55`

God bless....I hope the explanation is useful.