Question

Find the equation of the lines passing through the point of intersection lines 4x − y + 3 = 0 and 5x + 2y + 7 = 0, and through the point (−1, 2)

Answer 1

The equation of the straight line passing through the point of intersection of the lines.

4x – y + 3 = 0 and 5x + 2y + 7 = 0 is

(4x – y + 3) + λ(5x + 2y + 7) = 0   ......(1)

Through the point (–1, 2)

Given that line (1) passes through the point (–1, 2)

(1) ⇒ (4(–1) – 2 + 3) + λ(5(–1) + 2(2) + 7) = 0

(– 4 – 2 + 3) + λ(– 5 + 4 + 7) = 0

– 3 + 6λ = 0

⇒ λ = `3/6 = 1/2`

∴ The equation of the required line is

`(4x - y + 3) + 1/2 (5x + 2y + 7)` = 0

2(4x – y + 3) + (5x + 2y + 7) = 0

8x – 2y + 6 + 5x + 2y + 7 = 0

13x +13 = 0

⇒ x + 1 = 0