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

विकल्प

• 4p₁² + p₂² = a²

• p₁² + 4p₂² = a²

•  p₁² + p₂² = a²

• none of these

Answer 1

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.

\[p_1 = \left| \frac{- a}{\sqrt{\sec^2 \theta + + \cos e c^2 \theta}} \right| \text { and } p_2 = \left| \frac{- a\cos2\theta}{\sqrt{\cos^2 \theta + \sin^2 \theta}} \right|\]

\[ \Rightarrow p_1 = \left| \frac{- a\sin\theta\cos\theta}{\sqrt{\sin^2 \theta + \cos^2 \theta}} \right|\text { and } p_2 = \left| - a\cos2\theta \right|\]

\[ \Rightarrow p_1 = \frac{1}{2}\left| - a \times 2\sin\theta\cos\theta \right| \text { and } p_2 = \left| - a\cos2\theta \right|\]

\[ \Rightarrow p_1 = \frac{1}{2}\left| - a\sin2\theta \right| \text { and } p_2 = \left| - a\cos2\theta \right|\]

\[ \Rightarrow 4 {p_1}^2 + {p_2}^2 = a^2 \left( \sin^2 2\theta + \cos^2 2\theta \right)\]

\[ = a^2\]