Question

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

• A. 3
• B. 1
• C. 0
• D. 4
• E. 2

Answer 1

Answer: (D)

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