Question

Let the angle between the two curves $y=2\sqrt{x}$ and $x=2\sqrt{y}$ is $\theta$. If $\theta \ne \frac{\pi }{2}$ and it is acute. then $\sqrt{sin\theta +sin3\theta +sin5\theta }$ =

• A. $\frac{39}{25}$
• B. $\frac{39\sqrt{3}}{25}$
• C. $\frac{39}{25\sqrt{5}}$
• D. $\frac{39\sqrt{3}}{25\sqrt{5}}$

Answer 1

The correct option is D $\frac{39\sqrt{3}}{25\sqrt{5}}$

Given curves are
$y^2=4x,x^2=4y$
$P.I=(0,0),(4,4)$
At $(0,0),\theta =\frac{\pi }{2}$
At $(4,4),\theta =ta{n}^{-1}\frac{3}{4}\Rightarrow tan\theta =\frac{3}{4}$
$sin\theta +sin3\theta +sin5\theta =sin3\theta [2cos2\theta +1]$
$=\frac{117}{125}\times \frac{39}{25}$
$\sqrt{sin\theta +sin3\theta +sin5\theta }=\frac{39\sqrt{3}}{25\sqrt{5}}$