The equation of the pair of lines passing through origin and making an angle of 30⁰ with the line 3x – y – 1= 0 is

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$13 x^{2} + 12 x y + 3 y^{2} = 0$

$13 x^{2} - 12 x y + 3 y^{2} = 0$

$13 x^{2} + 12 x y - 3 y^{2} = 0$

$13 x^{2} - 12 x y - 3 y^{2} = 0$

answer is A.

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Detailed Solution

Given line 3x – y = 0

Comparing with lx + my + n = 0

$\begin{array}{l} l = 3, m = - 1, n = - 1 a n d α = 30 \\ Re q u i r e d E q u a t i o n i s {(l x + m y)}^{2} - {Tan}^{2} α {(m x - l y)}^{2} = 0 \\ \Rightarrow {(3 x - y)}^{2} - {Tan}^{2} 30 {(- x - 3 y)}^{2} = 0 \\ \Rightarrow {(3 x - y)}^{2} - \frac{1}{3} {(x + 3 y)}^{2} = 0 \\ \Rightarrow 3 (9 x^{2} + y^{2} - 6 x y) - (x^{2} + 9 y^{2} + 6 x y) = 0 \\ \Rightarrow 27 x^{2} + 3 y^{2} - 18 x y - x^{2} - 9 y^{2} - 6 x y = 0 \\ \Rightarrow 26 x^{2} - 24 x y - 6 y^{2} = 0 \\ \Rightarrow 13 x^{2} - 12 x y - 3 y^{2} = 0 \end{array}$