Question

Two lines passing through the point $(2,3)$ intersects each other at an angle of ${60}^{\circ }$. If slope of one line is $2$ , find equation of the other line.

Answer 1

$\tan \theta =\frac{m_1-m_2}{1+m_1{m}_2}$

$\Rightarrow \sqrt{3}=\frac{|2-m|}{1+2m}\Rightarrow \sqrt{3}+2\sqrt{3}m=12-ml$

$\Rightarrow (2\sqrt{3}+1)m=2-\sqrt{3}$ or $(2\sqrt{3}-1)m=2+\sqrt{3}$

$\Rightarrow m=\frac{2-\sqrt{3}}{2\sqrt{3}+1}$ or $\frac{2+\sqrt{3}}{2\sqrt{3}-1}$

equation of line $=(y-3)=\frac{2\pm \sqrt{3}}{2\sqrt{3}\mp 1}(x-2)$ ;

$\Rightarrow (2\sqrt{3}\mp 1)(y-3)=(2\pm \sqrt{3})(x-2)$

$\Rightarrow 2\sqrt{3}y-y-6\sqrt{3}+3=2x-4+\sqrt{3}x-2\sqrt{3}$ or $2\sqrt{3}y-6\sqrt{3}+y-3=2x-4-\sqrt{3}x+2\sqrt{3}$

$\Rightarrow (2\sqrt{3}-1)-(\sqrt{3}+2)x=-7+4\sqrt{3}$ or $(2\sqrt{3}+1)y+(\sqrt{3}-2)x=-1+8\sqrt{3}$