If `y=sqrt((a-x)(x-b))-(a-b)tan^(-1)(sqrt(a-x)/sqrt(x-b))` then `dy/dx=`

Options

A. `sqrt((a-x)(x-b))`
B. `(1)/(sqrt((a-x)(x-b)))`
C. `sqrt(((a-x)/(x-b)))`
D. `sqrt(((x-b)/(a-x)))`

Answer

C

Explanation

Let ` x = a cos^(2) theta + b sin^(2) theta `
` therefore a - x = (a - b) sin^(2) theta , x - b = (a - b) cos^(2) theta `
` therefore y = (a - b) sin theta cos theta - (a - b) theta `
` = ((a - b) sin 2 theta )/(2) - (a - b) theta `
` rArr (dy)/(d theta ) = (a - b) cos 2 theta - (a -b)`
` = - (a - b) sin^(2) theta - 2 (a - b) sin^(2) theta `
and `(dx)/(d theta ) = (b -a) sin 2 theta `
` therefore (dy)/(dx) = (-2(a -b)sin^(2) theta )/((b -a) sin^(2) theta )= tan theta = sqrt((a -x)/(x -b))`