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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
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Solution
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`
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