Question

The points P and Q have coordinates (-1, 6) and (9, 0) respectively.

The line l is perpendicular to PQ and passes through the mid-point of PQ.
Find an equation for l, giving your answer in the form ax + by + c =0, where a, b and c are integers.

Answer 1

`3y - 5x +11=0`

Explanation:

Given

`P(x_1,y_1) = (-1,6)`

`Q(x_2,y_2) = (9,0)`

Required

The equation of l

First, calculate the slope (m) of PQ

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (0-6)/(9--1)`

`m = (-6)/(10)`

`m = (-3)/(5)`

Since l is perpendicular to PQ, the slope of l is:

`m_2 = -(1)/(m)`

`m_2= -(1)/(-3/5)`

`m_2 = (5)/(3)`

Next, calculate the midpoint of PQ

`M = (1)/(2)(x_1 + x_2,y_1+y_2)`

`M = (1)/(2)(-1+9,6+0)`

`M = (1)/(2)(8,6)`

`M = (4,3)`

The equation of l is:

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

`y = (5)/(3)(x -4) +3`

Multiply through by 3

`3y = 5(x -4) +9`

Open bracket

`3y = 5x -20 +9`

`3y = 5x -11`

Rewrite as:

`3y - 5x +11=0`