The combined equation to a pair of straight lines passing through the origin and inclined at 45⁰, 60⁰ with x-axis is

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$\sqrt{3} x^{2} + (\sqrt{3} + 1) x y + y^{2} = 0$

$\sqrt{3} x^{2} - (\sqrt{3} + 1) x y - y^{2}$

$\sqrt{3} x^{2} - (\sqrt{3} + 1) x y + y^{2} = 0$

$\sqrt{3} x^{2} - (\sqrt{3} - 1) x y + y^{2}$

answer is C.

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Detailed Solution

$\begin{array}{l} L e t θ_{1} = 45^{0} \Rightarrow m_{1} = Tan θ_{1} = Tan 45^{0} = 1 \\ θ_{2} = 60^{0} \Rightarrow m_{2} = Tan θ_{2} = Tan 60^{0} = \sqrt{3} \\ ∴ E q u a t i o n o f l i n e s a r e y = m_{1} x, y = m_{2} x \end{array}$

$\begin{array}{l} \Rightarrow y = x, y = \sqrt{3} x \\ \Rightarrow x - y = 0, \sqrt{3} x - y = 0 \\ ∴ c o m b a i n e d E q u a t i o n i s (x - y) (\sqrt{3} x - y) = 0 \\ \Rightarrow \sqrt{3} x^{2} - (\sqrt{3} + 1) x y + y^{2} = 0 \end{array}$