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11. ⇒ (MHT CET 2021 20th September Morning Shift )

The joint equation of the pair of lines through the origin and making an equilateral triangle with the line $x = 3$ is

A. $3 x^{2} - y^{2} = 0$

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

C. $x^{2} - 3 y^{2} = 0$

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

Correct answer option is (C)

MHT CET 2021 20th September Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 28 English Explanation

Refer figure.

Slope of line $OA = \tan 30^{\circ} = \frac{1}{\sqrt{3}}$ and

Slope of line $OB = \tan (- 30^{\circ}) = \frac{- 1}{\sqrt{3}}$

$∴$ Equation of $OA$ is $y = \frac{1}{\sqrt{3}} x$ and equation of $OB$ is $y = \frac{- 1}{\sqrt{3}} x$

Hence required equation is $(x - \sqrt{3} y) (x + \sqrt{3} y) = 0$ i.e. $x^{2} - 3 y^{2} = 0$

⇒Pair of Straight Line