Question

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and  hence, find dy/dx if x=a cost, y=a sint

Answer 1

Given x=f(t),y=g(t) are differentiable function of parameter 't'

x=acost and y=asint

find dy/dx=?

x=acost

differentiate x w.r.t 't'

`dx/dt=d/dt(acost)`

`dx/dt=asint................(1)`

`y=asint`

`dy/dt=d/dt (asint)`

`dy/dt=-acost.............(2)`

dividing equation 2 by 1

`(dy/dt)/(dx/dt)=(-acost)/(asint)=-cost/sint......(3)`

`now " "x=acost`

`therefore cost=x/a`

`y=asint`

`therefore sintt=y/a`

from equation 3

`dy/dx=-(x/a)/(y/a)=-x/y`