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 $m=tan$($\frac{2\pi}{3}$)=$-\sqrt{3}$

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, $\sqrt3x+y-2=0$

The slope of line parallel to line $\sqrt3x+y-2=0$ is $-\sqrt{3}$
It is given that the line parallel to line $\sqrt3x+y-2=0$ 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 the point (0,2) and having a slope $-\sqrt{3}$ is
$y-\left(-2\right)=-\sqrt{3}\left(x-0\right)$
$y+2=-\sqrt3x$
$\sqrt3x+y+2=0$