Question

If $y=\sqrt{(a-x)(x-b)}-(a-b){\tan }^{-1}\sqrt{\frac{a-x}{x-b}}$, then $\frac{dy}{dx}$ is equal to

• A. $1$
• B. $\sqrt{\frac{a-x}{x-b}}$
• C. $\sqrt{(a-x)(x-b)}$
• D. $\frac{1}{\sqrt{(a-x)(b-x)}}$

Answer 1

The correct option is B $\sqrt{\frac{a-x}{x-b}}$

Let $x=a{\cos }^2\theta +b{\sin }^2\theta$
Here $a-x=a-a{\cos }^2\theta -b{\sin }^2\theta =(a-b){\sin }^2\theta$
and $x-b=a{\cos }^2\theta -b{\sin }^2\theta -b=(a-b){\cos }^2\theta$
and $y=(a-b)\sin \theta \cdot \cos \theta -(a-b){\tan }^{-1}\tan \theta$
$=\frac{a-b}{2}\sin 2\theta -(a-b)\theta$
Therefore, $\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=\frac{(a-b)\cos 2\theta -(a-b)}{(b-a)\sin 2\theta }$
$=\frac{1-\cos 2\theta }{\sin \theta }=\tan \theta =\sqrt{\frac{a-x}{x-b}}$