# Find equation of the line through the point (0, 2) making an angle

Question:

Find equation of the line through the point $(0,2)$ making an angle $\frac{2 \pi}{3}$ with the positive $x$-axis. Also, find the equation of line parallel to it and crossing the $y$-axis at a distance of 2 units below the origin.

Solution:

The slope of the line making an angle $\frac{2 \pi}{3}$ with the positive $x$-axis is $m=\tan \left(\frac{2 \pi}{3}\right)=-\sqrt{3}$

Now, the equation of the line passing through point $(0,2)$ and having a slope $-\sqrt{3}$ is $(y-2)=-\sqrt{3}(x-0)$.

$y-2=-\sqrt{3} x$

i.e., $\sqrt{3} x+y-2=0$

The slope of line parallel to line $\sqrt{3} x+y-2=0$ is $-\sqrt{3}$.

It is given that the line parallel to line $\sqrt{3} x+y-2=0$ crosses the $y$-axis 2 units below the origin i.e., it passes through point ( $0,-2$ ).

Hence, the equation of the line passing through point $(0,-2)$ and having a slope $-\sqrt{3}$ is

$y-(-2)=-\sqrt{3}(x-0)$

$y+2=-\sqrt{3} x$

$\sqrt{3} x+y+2=0$