Question

If $P_1$ and $P_2$ be the length of perpendiculars from the origin upon the straight lines $x\sec \theta +ycosec\theta =a$ and $x\cos \theta -y\sin \theta =a\cos 2\theta$ respectively, then the value of $4P_1{}^2+P_2{}^2$.

• A. $a^2$
• B. $2a^2$
• C. $a^2/2$
• D. $3a^2$

Answer 1

Answer: (A)

We have $P_1$ = length of perpendicular from $(0,0)$ on
$x\sec \theta +y\cos ec\theta =a$
i.e. $P_1=\left|\frac{a}{\sqrt{{\sec }^2\theta +\cos e{c}^2\theta }}\right|=\left|a\sin \theta \cos \theta \right|$
$=\left|\frac{a}{2}\sin 2\theta \right|2P_1=\left|a\sin 2\theta \right|$
$P_2$= Length of the perpendicular from $(0,0)$ on
$x\cos \theta -y\sin \theta =a\cos 2\theta$
$P_2=\left|\frac{a\cos 2\theta }{\sqrt{{\cos }^2\theta +{\sin }^2\theta }}\right|=\left|a\cos 2\theta \right|$
Now, $4P_1^2+P_2^2=a^2{\sin }^{2}2\theta +a^2{\cos }^{2}2\theta =a^2.$