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Question
Find the equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line makes an angle tan−1 \[\left( \frac{5}{12} \right)\] with the positive direction of x-axi .
Solution
Here, p = 3,
\[\alpha = \tan^{- 1} \left( \frac{5}{12} \right)\]
\[\therefore\text { tan }\alpha = \frac{5}{12}\]
\[ \Rightarrow \text { sin} \alpha = \frac{5}{13} \text { and } cos\alpha = \frac{12}{13}\]
So, the equation of the line in normal form is
\[x\text { cos }\alpha + y\text { sin }\alpha = p\]
\[ \Rightarrow \frac{12x}{13} + \frac{5y}{13} = 3\]
\[ \Rightarrow 12x + 5y = 39\]
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