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If x=e^(theta)(sintheta-costheta),y=e^(t...

If `x=e^(theta)(sintheta-costheta),y=e^(theta)(sintheta+costheta)`, then `(dy)/(dx)` at `theta=(pi)/(4)` is

A

`1`

B

`0`

C

`1/sqrt(2)`

D

`sqrt(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
`x=e^(theta)(sintheta-costheta),y=e^(theta)(sintheta+costheta)`,
`therefore(dx)/(d theta)=(d)/(d theta)[e^theta(sin theta-costheta)]`
`=e^(theta).(d)/(d theta)(sin theta-cos theta)+(sintheta-costheta).(d)/(d theta)(e^(theta))`
`=e^(theta)(costheta+sintheta+sintheta-costheta)`
`=2e^(theta)sintheta`
`and(dy)/(d theta)=(d)/(d theta)[e^(theta)(sintheta+costheta)]`
`=e^(theta).(d)/(d theta)(sintheta+costheta)+(sintheta+costheta).(d)/(d theta)(e^(theta))`
`=e^(theta)(costheta-sintheta+sintheta+costheta)`
`=2e^(theta)costheta`
`therefore(dy)/(dx)=((dy//d theta))/((dx//d theta))=(2e^(theta)costheta)/(2e^(theta)sin theta)=cottheta`
`therefore((dy)/(dx))_(at theta=pi//4)=cot.(pi)/(4)=1`
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NAVNEET PUBLICATION - MAHARASHTRA BOARD-DIFFERENTIATION-Ex. 23 Solved Examples
  1. If x=e^(theta)(sintheta-costheta),y=e^(theta)(sintheta+costheta), then...

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