Question

If $p_1$ and $p_2$ are the lengths of the perpendiculars from the origin upon the lines $xsec\theta +ycosec\theta =a$ and $xcos\theta -ysin\theta =acos2\theta$ respectively, then

• A. $4p_1^2+p_2^2=a^2$
• B. $p_1^2+4p_2^2=a^2$
• C. $p_1^2+p_2^2=a^2$
• D. none of these

Answer 1

The correct option is A

$4p_1^2+p_2^2=a^2$

The given lines are

$xsec\theta +ycosec\theta =a...\left(1\right)$

$xcos\theta -ysin\theta =acos2\theta ...\left(2\right)$

$p_1$ and $p_2$ are the perpendicular from the origin upon the lines (1) and (2), respectively.

$\Rightarrow p_1=\left|\frac{-a}{\sqrt{se{c}^2\theta +cose{c}^2\theta }}\right|$ and $p_2=\left|\frac{-acos2\theta }{\sqrt{co{s}^2\theta }+si{n}^2\theta }\right|$

$\Rightarrow p_1=\left|\frac{asin\theta cos\theta }{\sqrt{si{n}^2\theta +co{s}^2\theta }}\right|$ and $p_2=|-acos2\theta |$

$\Rightarrow p_1=\frac{1}{2}|-a\times 2sin\theta cos\theta |$ and $p_2=|-acos2\theta |$

$\Rightarrow 4p_1^2+p_2^2=a^2(si{n}^{2}2\theta +co{s}^{2}2\theta )=a^2$