Question

If $p$ and $p^{\prime }$ be the perpendiculars from the origin upon the straight lines $xsec\theta +ycosec\theta =a$ and $xcos\theta -ysin\theta =acos2\theta$ respectively, then the value of the expression $4p^2+p^{\prime 2}$ is

• A. $a^2$
• B. $3a^2$
• C. $2a^2$
• D. $4a^2$

Answer 1

Answer: (A)

We have $p=$ length of the perpendicular from $(0,0)$ on $xsec\theta +ycosec\theta =a$
i.e., $p=\frac{a}{\sqrt{se{c}^2\theta +cose{c}^2\theta }}=\frac{asin2\theta }{2}$
$\Rightarrow 2p=asin2\theta \dots \left(i\right)$
$p^{\prime }=$ length of the perpendicular from $\left(0,0\right)$ on
$xcos\theta -ysin\theta =acos2\theta$
i.e., $p^{\prime }=\frac{acos2\theta }{\sqrt{co{s}^2\theta +si{n}^2\theta }}=acos2\theta \dots \left(ii\right)$
Squaring and adding $\left(i\right)$ and $\left(ii\right)$, we get
$4p^2+p^{\prime 2}=a^2$