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Equation of line in Straight lines

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Question

A straight line L through the point $(3, - 2)$ is inclined at an angle $60 °$ to the line $\sqrt{3} x + y = 1 .$ If $L$ If L also intersects the x-axis, then the equation of $L$ is

Moderate

Solution

Let the slope of $L$ be $m$
then, $|\frac{m + \sqrt{3}}{1 - \sqrt{3 m}}| = \tan 60^{\circ} = \sqrt{3}$
$\Rightarrow m + \sqrt{3} = \sqrt{3} (1 - \sqrt{3} m)$
or $m + \sqrt{3} = - \sqrt{3} (1 - \sqrt{3} m)$
$\Rightarrow m = 0$ or $n = \sqrt{3}$
But $n \neq 0$ as $L$ intersects $x$ - axis
Hence equation of L is
$y + 2 = \sqrt{3} (x - 3)$
or $y - \sqrt{3} x + 2 + 3 \sqrt{3} = 0$

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