If `x=a(1+costheta)` , `y=a(theta+sintheta)` , prove that `(d^2y)/(dx^2)=(-1)/a` at `theta=pi/2` .

Options

A. `-(1)/(a)`
B. `(1)/(a)`
C. `-1`
D. `-2`

Answer

A

Explanation

Given , ` x = a (1 + cos theta) , y = a (a theta + sin theta)`
` rArr (dx)/(d theta) = - a sin theta , (dy)/(d theta ) = a (1 + cos theta)`
` therefore (dy)/(dx) = (1 + cos theta)/( - sin theta) = (2 cos^(2) ""(theta)/(2))/(-2 sin "" (theta)/(2) cos "" (theta)/(2))`
` rArr (dy)/(dx) =- cot"" (theta)/(2)`
` therefore (d^(2)y)/(dx^(2)) = (d)/(dx) ((dy)/(dx))= (d)/(d theta)((dy)/(dx)) . (d theta)/(dx)`
` = (d)/(d theta) (- cot "" ( theta)/(2)). (1) /(- a sin theta) `
` = (1)/(2) "cosec"^(2) (theta)/(2). (1)/( - a sin theta)`
`therefore ((d^(2) y)/(dx^(2)))_(e = (pi)/(2)) = (1)/(2) . 2 . (1)/(-a) =- (1)/(a)`