# Q : 12     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.

Let the slope of two lines are $m_1 \ and \ m_2$   respectively
It is given the lines intersects each other at an angle of  $60^{\circ}$  and slope of the line is 2
Now,
$m_1 = m\ and \ m_2= 2 \ and \ \theta = 60\degree$
$\tan \theta = \left | \frac{m_2-m_1}{1+m_1m_2} \right |$
$\tan 60\degree = \left | \frac{2-m}{1+2m} \right |$
$\sqrt3 = \left | \frac{2-m}{1+2m} \right |$
$\sqrt3 = \frac{2-m}{1+2m} \ \ \ \ \ \ or \ \ \ \ \ \ \ \ \ \sqrt 3 = -\left ( \frac{2-m}{1+2m} \right )$
$m = \frac{2-\sqrt3}{2\sqrt3+1} \ \ \ \ \ \ or \ \ \ \ \ \ \ \ \ \ m = \frac{-(2+\sqrt3)}{2\sqrt3-1}$
Now, the equation of line passing through point (2 ,3) and with slope  $\frac{2-\sqrt3}{2\sqrt3+1}$  is
$(y-3)= \frac{2-\sqrt3}{2\sqrt3+1}(x-2)$
$(2\sqrt3+1)(y-3)=(2-\sqrt3)(x-2)$
$x(\sqrt3-2)+y(2\sqrt3+1)=-1+8\sqrt3$                        -(i)

Similarly,
Now , equation of line passing through point (2 ,3) and with slope  $\frac{-(2+\sqrt3)}{2\sqrt3-1}$  is
$(y-3)=\frac{-(2+\sqrt3)}{2\sqrt3-1}(x-2)$
$(2\sqrt3-1)(y-3)= -(2+\sqrt3)(x-2)$
$x(2+\sqrt3)+y(2\sqrt3-1)=1+8\sqrt3$                             -(ii)

Therefore, equation of line is    $x(\sqrt3-2)+y(2\sqrt3+1)=-1+8\sqrt3$     or      $x(2+\sqrt3)+y(2\sqrt3-1)=1+8\sqrt3$

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