Question

If $x=5+2\sec \theta$ and $y=5+2\tan \theta$, then ${(x-5)}^2-{(y-5)}^2$ is equal to

• A. $3$
• B. $1$
• C. $0$
• D. $4$
• E. $2$

Answer 1

Answer: (D)

$4$

Given, $x=5+2\sec \theta$
$\Rightarrow x-5=2\sec \theta$ ...(i)
and $y=5+2\tan \theta$
$\Rightarrow y-5=2\tan \theta$ ....(ii)
From equations (i) and (ii), we get
${(x-5)}^2-{(y-5)}^2$ $={\left(2\sec \theta \right)}^2-{\left(2\tan \theta \right)}^2$
$=4{\sec }^2\theta -4{\tan }^2\theta$

$=4({\sec }^2\theta -{\tan }^2\theta )=4$