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. `(a)/(sqrt(2))`
D. `sqrt(2)a`

Answer

D

Explanation

Given curves are
`x=a(theta+sintheta),y=a(1-costheta)`
`:.(dy)/(dx)=(dy//d theta)/(dx//d theta)=(a(sintheta))/(a(1+costheta))`
`=(2sin""(theta)/(2)cos""(theta)/(2))/(2cos^(2)""(theta)/(2))=tan""(theta)/(2)`
`implies((dy)/(dx))_((theta=(pi)/(2)))=tan""(pi)/(4)=1`
At `" "theta=(pi)/(2),y=a(1-cos""(pi)/(2))=a`
`:.` Length of normal
`=ysqrt(1+((dy)/(dx))^(2))=asqrt(1+(1)^(2))=sqrt(2)a`