Combined equation of pair of lines passing through origin and inclined at 300 and 600 respectively with x-axis is

Let θ1=300 and θ2=600 ⇒ m1=Tan300⇒ 13, m2=Tan600=3Equation of lines passing through 0, 0 is y=m1x1, y=m2x⇒ y=x3,∴ y=3x⇒ x−3y=0, 3x−y=0∴ Combainded Equation is x−3y3x−y=0⇒ 3x2−4xy+3y2=0

Combined equation of pair of lines passing through origin and inclined at 300 and 600 respectively with x-axis is

Let θ1=300 and θ2=600 ⇒ m1=Tan300⇒ 13, m2=Tan600=3Equation of lines passing through 0, 0 is y=m1x1, y=m2x⇒ y=x3,∴ y=3x⇒ x−3y=0, 3x−y=0∴ Combainded Equation is x−3y3x−y=0⇒ 3x2−4xy+3y2=0

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Combined equation of pair of lines passing through origin and inclined at 30⁰ and 60⁰ respectively with x-axis is

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$y^{2} - \sqrt{3} x^{2} = 0$

$\sqrt{3} y^{2} + 4 x y - \sqrt{3} x^{2} = 0$

$\sqrt{3} y^{2} - 4 x y + \sqrt{3} x^{2} = 0$

$\sqrt{3} y^{2} - x^{2} = 0$

answer is D.

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Detailed Solution

$\begin{array}{l} L e t θ_{1} = 30^{0} a n d θ_{2} = 60^{0} \Rightarrow m_{1} = Tan 30^{0} \Rightarrow \frac{1}{\sqrt{3}}, m_{2} = Tan 60^{0} = \sqrt{3} \\ E q u a t i o n o f l i n e s p a s \sin g t h r o u g h (0, 0) i s y = m_{1} x_{1}, y = m_{2} x \\ \Rightarrow y = \frac{x}{\sqrt{3}}, ∴ y = \sqrt{3} x \\ \Rightarrow x - \sqrt{3} y = 0, \sqrt{3} x - y = 0 \\ ∴ C o m b a i n d e d E q u a t i o n i s (x - \sqrt{3} y) (\sqrt{3} x - y) = 0 \\ \Rightarrow \sqrt{3} x^{2} - 4 x y + \sqrt{3} y^{2} = 0 \end{array}$