Question

Write the equation of the line whose perpendicular distance from the origin to the line is $6$ unit and angle made by the perpendicular with $+ve$ axis is ${30}^0$

Answer 1

Normal equation of line-

$x\cos \theta +y\sin \theta =p$

where,

$p=\perp \text{distance from the origin}=6$

$\theta =\text{angle which makes by the normal (perpendicular) with positive axis}=30°$

$\therefore$ Equation of line-

$x\cos 30°+y\sin 30°=6$

$\Rightarrow \frac{\sqrt{3}}{2}x+\frac{1}{2}y=6\{\because \cos 30°=\frac{\sqrt{3}}{2}\&\sin 30°=\frac{1}{2}\}$

$\Rightarrow \sqrt{3}x+y=12$

Hence, the equation of line is $\sqrt{3}x+y=12$.