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, 

\[\text { sin }\alpha = \frac{1}{3}\]

\[\therefore \text { cos}\alpha = \sqrt{1 - \sin^2 \alpha}\]

\[ \Rightarrow\text {  cos }\alpha = \sqrt{1 - \frac{1}{9}} = \frac{2\sqrt{2}}{3}\]

So, the equation of the line in normal form is

\[x\text { cos }\alpha + y\text { sin }\alpha = p\]

\[ \Rightarrow \frac{2\sqrt{2}x}{3} + \frac{y}{3} = 2\]

\[ \Rightarrow 2\sqrt{2}x + y = 6\]