Question

Find the equation of the line perpendicular distance from the origin is $5$ units and the angle made by the perpendicular with the positive $x$-axis is $30{}^{\circ }$.

Answer 1

We know that, If $p$ is the length of the normal from origin to a line and $w$ is the angle made by the normal with the positive direction of $x$-axis then the equation of the line is given by $x\cos w+y\sin w=p$.

Here, $p=5$ units and $w={30}^{\circ }$

Thus, the required equation of the given line is

$x\cos {30}^{\circ }+y\sin {30}^{\circ }=5$

$\Rightarrow x\frac{\sqrt{3}}{2}+y.\frac{1}{2}=5$

$\Rightarrow \sqrt{3}x+y=10$