Question

If $p$ and $p^{\prime }$ are the perpendiculars from the origin upon the lines $x\sec \theta +y\text{csc}\theta =a$ and $x\cos \theta -y\sin \theta =a\cos 2\theta$ respectively then

• A. $4p^2+p^{{}^{\prime }2}=a^2$
• B. $p^2+4p^{{}^{\prime }2}=a^2$
• C. $p^2+p^{{}^{\prime }2}=a^2$
• D. $noneofthese$

Answer 1

The correct option is A $4p^2+p^{{}^{\prime }2}=a^2$

Perpendiculars are drawn from $(0,0)$ onto the line $x\sec \theta +y\text{cosec}\theta -\alpha =0$
$p=\frac{|-a|}{\sqrt{{\sec }^2\theta +{\text{cosec}}^2\theta }}=|-a\sin \theta \cos \theta |$
$p^{\prime }=\frac{|-a\cos 2\theta |}{\sqrt{{\cos }^2\theta +{\sin }^2\theta }}=|-a\cos 2\theta |$

$4p^2+p^{{}^{\prime }2}=a^2$