Question

Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.

Answer 1

Given: p = 4 and ω = 30°.

Equation of the line in normal form is

$$\begin{gathered} x\cos \omega +y\sin \omega =p \\ \Rightarrow x\cos \left({30}^{°}\right)+y\sin \left({30}^{°}\right)=4 \\ \Rightarrow x\frac{\sqrt{3}}{2}+y\frac{1}{2}=4 \\ \Rightarrow \sqrt{3}x+y=8 \end{gathered}$$

Hence, the equation of the line is $\sqrt{3}x+y=8$.