Question

Find ‘a’ if A(2a + 2, 3), B(7, 4) and C(2a + 5, 2) are collinear.

Answer 1

Given:

• A = (x₁, y₁) = (2a + 2, 3)
• B = (x₂, y₂) = (7, 4)
• C = (x₃, y₃) = (2a + 5, 2)

Points A, B, and C are collinear if: Slope of AB = Slope of BC

The formula for slope (m) is:

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

⇒ Calculate the Slope of AB:

Slope of AB =`(4 - 3)/(7 - (2a + 2))`

Slope of AB = `1/(7 - 2a - 2)`

∴ Slope of AB = `1/(5 - 2a)`

⇒ Calculate the Slope of AB:

Slope of BC = `(2 - 4)/((2a + 5) - 7)`

Slope of BC = `(-2)/(2a - 2)`

Equate the slopes and solve for a:

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

Cross-multiply the terms:

1(2a − 2) = −2(5 − 2a)

2a − 2 = −10 + 4a

Rearrange the terms to solve for a:

10 − 2 = 4a − 2a

8 = 2a

a = `8/2`

∴ a = 4

Hence, the value of a is 4.