If `x=(1-t^(2))/(1+t^(2))` and `y=(2t)/(1+t^(2))`, then `(dy)/(dx)` is equal to

Options

A. `-(y)/(x)`
B. `(y)/(x)`
C. `-(x)/(y)`
D. `(x)/(y)`

Answer

C

Explanation

Given ` x = (1-t^(2))/(1+t^(2))`
and ` y = (2t)/(1+t^(2))`
Put `t = tan theta` in both the equations,
we get
`x = (1- tan^(2) theta)/(1+ tan^(2) theta) = cos 2 theta " ".....(i)`
and ` y = (2 tan theta)/(1+tan^(2) theta) = sin 2 theta " ".....(ii)`
On differenting both the Eqs. (i)
and (ii) , we get
`(dx)/(d theta) = - 2 sin 2 theta`
and `(dy)/(d theta) = 2 cos 2 theta`
`therefore " "(dy)/(dx) = ((dy)/(d theta))/((dx)/(d theta))`
`= - (cos 2 theta)/(sin 2 theta) = - (x)/(y)`