Question

The length of the perpendicular from the origin to a line is 7 and the line makes an angle of ${150}^{\circ }$ with the positive direction of y-axis. Find the equation of the line.

Answer 1

$\text{Here, P = perpendicular distance from origin = 7}$

$\text{Angle made with y-axis is}{150}^{\circ },$

$\therefore \text{Angle made with x-axis is}{30}^{\circ }$

$cos\alpha =cos{30}^{\circ }=\frac{\sqrt{3}}{2}$

$sin\alpha =sin{30}^{\circ }=\frac{1}{2}$

$\text{The equation of line is}$

$xcos\alpha +ysin\alpha =p$

$x\left(\frac{\sqrt{3}}{2}\right)+y\left(\frac{1}{2}\right)=7$

$\sqrt{3}x+y=14$