Q : 12 Two lines passing through the point intersects each other at an angle of . If slope of one line is , find equation of the other line.

Name: Two lines passing through the point (2, 3) intersects each other at an angle of 60 degree. If slope of one line is 2, find equation of the other line.
Uploaded: Thu, 2019-06-13 09:55
Description: Two lines passing through the point (2, 3) intersects each other at an angle of 60 degree. If slope of one line is 2, find equation of the other line.

Two lines passing through the point (2, 3) intersects each other at an angle of 60 degree. If slope of one line is 2, find equation of the other line.

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.

Answers (1)

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    $x(2+\sqrt3)+y(2\sqrt3-1)=1+8\sqrt3$