Let the equation of a curve be `x=a(theta+sin theta),y=a(1-cos theta)`. If `theta` changes at a constant rate k then the rate of change of the slope of the tangent to the curve at `theta=pi/3` is (a) `(2k)/sqrt3` (b) `k/sqrt3` (c) k (d) none of these

Options

A. `2k//sqrt3`
B. `k//sqrt3`
C. k
D. none of these

Answer

D

Explanation

`(dy)/(dx)=((dy)/(d theta))/((dx)/(d theta))=(1)/(2)sec^(2).(theta)/(2).(d theta)/(dt)=(k)/(2)sec^(2).(theta)/(2)`
`therefore" Required rate"=(k)/(2).sec^(2).(pi)/(6)=(k)/(2).((2)/(sqrt3))^(2)`