Question

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

Answer 1

Since the points A(a, 3), B(2, 1) and C(5, a) are collinear,
slope of AB = slope of BC

`(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)`

`=>a^2 - 3a + 2 = 6`

`=> a^2 - 3a -  4= 0`

`=> a = -1 or 4`

When a = 4, we have A (4,3) and B (2,1)

∴ Slope of AB = `(1-3)/(2 -4) = (-2)/(-2) = 1`

And, equation of line is given by

y - 3 = 1(x -4)

`=> y - 3 = x - 4`

`=> x - y = 1`

When a = -1, we have A ( -1,3) and B (2,1)

∴ Slope of AB = `(1-3)/(2 +1) = (-2)/3`

And, equation of line is given by

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

`=> 3y - 9 = -2x - 2`

`=> 2x + 3y = 7`