Assuming:

P₁ P₂ be the intercept between the lines 5x − y − 4 = 0 and 3x + 4y − 4 = 0.

P₁ P₂ makes an angle θ with the positive x–axis.

Given: (x₁, y₁) = A (1, 5)

Explanation:

So, the equation of the line passing through A (1, 5) is

Formula Used:

Let AP₁ = AP₂ = r

Then, the coordinates of P1 and P2 are given by

and

So, the coordinates of P₁ and P₂ are 1 + rcosθ, 5 + rsinθ and 1 – rcosθ, 5 – rsinθ, respectively.

Clearly, P₁ and P₂ lie on 5x − y − 4 = 0 and 3x + 4y − 4 = 0, respectively.

∴5(1 + rcos θ) – 5 – rsin θ – 4 = 0 and 3(1 – r cos θ) + 4(5 – rsin θ) + 4(5 – r sin θ) – 4 = 0

and

⇒ 95 cos θ – 19 sinθ = 12 cos + 16 sinθ

⇒ 83 cosθ = 35 sinθ

⇒ tan θ = 83/35

Thus, the equation of the required line is

⇒ 83x – 35y + 92 = 0

Hence, the equation of line is 83x – 35 y + 92 = 0