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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)
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Solution
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
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