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find the equation of the lines through the point of intersection of the lines x-3y+1=0 and 2x+5y-9=0 whose distance from the origin is sqrt(5)

$let required line be (x - 3 y + 1) + λ (2 x + 5 y - 9) = 0 - - - - - - (i) \Rightarrow (1 + 2 λ) x + (- 3 + 5 λ) y \pm (1 - 9 λ) = 0 Now the distance of the line from the origin be P= \frac{| - c |}{\sqrt{a^{2} + b^{2}}} \Rightarrow \sqrt{5} = \frac{| 1 - 9 λ |}{\sqrt{(1 + 2 λ)^{2} + (5 λ - 3)^{2}}} \Rightarrow 5 [(1 + 2 λ)^{2} + (5 λ - 3)^{2}] = (1 - 9 λ)^{2} (∵ Squaring on both sides) \Rightarrow 64 λ^{2} - 112 λ + 49 = 0 \Rightarrow (8 λ - 7)^{2} = 0 \Rightarrow λ = \frac{7}{8} Now substitute λ = \frac{7}{8} in eq(i) \Rightarrow (x - 3 y + 1) + \frac{7}{8} (2 x + 5 y - 9) = 0 \Rightarrow 8 x - 24 y + 8 + 14 x + 35 y - 63 = 0 \Rightarrow 22 x + 11 y - 55 = 0 \Rightarrow 2 x + y - 5 = 0 Is the required line$

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