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.

Answer 1

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

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

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

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

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

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

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