Question

If p₁ and p₂ are the lengths of the perpendiculars from the origin upon the lines x sec θ + y cosec θ = a and x cos θ − y sin θ = a cos 2 θ respectively, then
(a) 4p₁² + p₂² = a²
(b) p₁² + 4p₂² = a²
(c) p₁² + p₂² = a²
(d) none of these

Answer 1

(a) 4p₁² +p₂² = a²

The given lines are

x sec θ + y cosec θ = a ... (1)

x cos θ − y sin θ = a cos 2 θ ... (2)

p₁ and p₂ are the perpendiculars from the origin upon the lines (1) and (2), respectively.

$$\begin{gathered} p_1=\left|\frac{-a}{\sqrt{{\sec }^2\theta ++\cos \mathrm{e}{\mathrm{c}}^2\theta }}\right|\mathrm{and}{p}_2=\left|\frac{-a\cos 2\theta }{\sqrt{{\cos }^2\theta +{\sin }^2\theta }}\right| \\ \Rightarrow p_1=\left|\frac{-a\sin \theta \cos \theta }{\sqrt{{\sin }^2\theta +{\cos }^2\theta }}\right|\mathrm{and}{p}_2=\left|-a\cos 2\theta \right| \\ \Rightarrow p_1=\frac{1}{2}\left|-a\times 2\sin \theta \cos \theta \right|\mathrm{and}{p}_2=\left|-a\cos 2\theta \right| \\ \Rightarrow p_1=\frac{1}{2}\left|-a\sin 2\theta \right|\mathrm{and}{p}_2=\left|-a\cos 2\theta \right| \\ \Rightarrow 4{p_1}^2+{p_2}^2=a^2\left({\sin }^{2}2\theta +{\cos }^{2}2\theta \right) \\ =a^2 \end{gathered}$$