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-axis.

Answer 1

Here, p = 3, $\mathrm{\alpha }={\tan }^{-1}\left(\frac{5}{12}\right)$

$$\begin{gathered} \therefore \mathrm{tan\alpha }=\frac{5}{12} \\ \Rightarrow \mathrm{sin\alpha }=\frac{5}{13}\mathrm{and}\mathrm{cos\alpha }=\frac{12}{13} \end{gathered}$$

So, the equation of the line in normal form is

$$\begin{gathered} x\mathrm{cos\alpha }+y\mathrm{sin\alpha }=p \\ \Rightarrow \frac{12x}{13}+\frac{5y}{13}=3 \\ \Rightarrow 12x+5y=39 \end{gathered}$$