If ST and SN are the lengths of the subtangent and the subnormal at the point `theta=(pi)/2` on the curve `x=a(theta+sin theta),y=a(1-cos theta),a!=1` then

Options

A. `ST=SN`
B. `ST=2SN`
C. `ST^(2)=aSN^(3)`
D. `ST^(3)=aSN`

Answer

A

Explanation

Given `x=a (theta+sin theta)` and `y=a(1- cos theta)`
`implies(dx)/(d theta)=a (1+cos theta) `and `(dy)/(d theta) = a sin theta`
`:. (dy)/(dx)=(a sin theta)/(a(1+cos theta))= "tan" (theta)/2`
Now length of subtangent `= |y/(dy//dx)|`
`impliesST=(a(1-cos theta))/(tan (theta//2))=a sin theta`
Length oif subtangent at `theta=(pi)/2`,
`ST=a "sin"(pi)/2=a`
and length of subnormal `=|y (dy)/(dx)|`
`impliesSN=a(1-cos theta). "tan"(theta)/2=a.2 "sin"^(2)(theta)/2"tan"(theta)/2`
Length of subnormal at `thea=(pi)/2`
`SN=a.2 . 1/2=a`
Hence `SN=ST`