Find the length of normal to the curve `x=a(theta+sintheta),y=a(1-costheta)` at `theta=pi/2dot`

Options

A. `2a`
B. `a/2`
C. `2/(sqrt(2))`
D. `sqrt(2)a`

Answer

D

Explanation

Given parametric equations of curve are
`x=a(theta+ sin theta),y=a(1-cos theta)`
`:.(dy)/(dx)=((dy)/(d theta))/((dx)/(d theta))=(a (sin theta))/(a(1+cos theta))=(2 "sin" (theta)/2 "cos"(theta)/2)/(2 "cos"^(2)(theta)/2)="tan"(theta)/2`
At `theta=(pi)/2,((dy)/(dx))_((theta=(pi)/2))="tan"(pi)/4=1`
`:.` Length of normal `=y sqrt(1+((dy)/(dx))^(2))`
`=a(1-"cos"(pi)/2)sqrt(1+(1)^(2))=sqrt(2)a`