Question

If \(P_1\) and \(P_2\) be the lengths of perpendiculars from the origin upon the straight lines \(x\sec\theta+y\cosec\theta=a\) and \(x\cos\theta-y\sin\theta=a\cos2\theta\) respectively, then the value of \(4P_1^2+P_2^2\) is:

• A. \(a^2\)
• B. \(2a^2\)
• C. \(\dfrac{a^2}{2}\)
• D. \(3a^2\)

Hint:

Always reduce the line to standard form before using distance formula.

Answer 1

The Correct Option is D

Step 1: Perpendicular distance from origin to \(Ax+By+C=0\) is \(|C|/\sqrt{A^2+B^2}\).
Step 2: Compute \(P_1\) and \(P_2\) using the given equations.
Step 3: Substitution gives: \[ 4P_1^2+P_2^2=3a^2 \]