Question

Find the equation of the straight line on which the length of the perpendicular from the origin is 2 and the perpendicular makes an angle α with x-axis such that sin α = $\frac{1}{3}$.

Answer 1

Here, p = 2, $\mathrm{sin\alpha }=\frac{1}{3}$

$$\begin{gathered} \therefore \mathrm{cos\alpha }=\sqrt{1-{\sin }^2\mathrm{\alpha }} \\ \Rightarrow \mathrm{cos\alpha }=\sqrt{1-\frac{1}{9}}=\frac{2\sqrt{2}}{3} \end{gathered}$$

So, the equation of the line in normal form is

$$\begin{gathered} x\mathrm{cos\alpha }+y\mathrm{sin\alpha }=p \\ \Rightarrow \frac{2\sqrt{2}x}{3}+\frac{y}{3}=2 \\ \Rightarrow 2\sqrt{2}x+y=6 \end{gathered}$$