Question

Find the value of a for which the points A(a, 3), B(2, 1) and C(5, a) are collinear. Hence, find the equation of the line.

Answer 1

If 3 points are collinear, the slope between any 2 points is the same.

Thus, for A(a, 3), B(2, 1) and C(5, a) to be collinear, the slope Between A and B and between B and C should be the same. 

`=> (1 - 3)/(2 - a) = (a - 1)/(5 - 2)`

`=> (-2)/(2 - a) = (a - 1)/3`

`=> 2/(a - 2) = (a - 1)/3`

`=>` 6 = (a – 2)(a – 1)

`=>` a² – 3a + 2 = 6

`=>` a² – 3a – 4 = 0

`=>` a = –1 or 4

Thus, slope can be:

`2/(a - 2) = 2/(-1 -2) = -2/3` OR `2/(a - 2) = 2/(4 - 2) = 1`

Thus, the equation of the line can be:

`(y - 1)/(x -2) = - 2/3`

Equation of line

`=>` y – y₁ = m(x – x₁)

`=> y - 3 = (-2)/3[x - (-1)]`

`=>` 3(y – 3) = –2(x + 1)

`=>` 3y – 9 = –2x – 2

`=>` 2x + 3y = –2 + 9

`=>` 2x + 3y = 7

or 

`(y - 1)/(x - 2) = 1`

`=>` y – x = –1

`=>` x – y = 1