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Question

10.x= a (cos θ + θ sin θ), ya (sin θ-θ cos θ)

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Solution

It is given that x and y are parametrically connected by the equations,

x=a( cosθ+θsinθ )(1)

And,

y=a( sinθθcosθ )(2)

Differentiate equation (2) with respect to θ.

dy dθ =a d( cosθ+θsinθ ) dθ dy dθ =a( cosθcosθ+θsinθ ) dy dθ =a( θsinθ ) dy dθ =aθsinθ

Differentiate equation (1) with respect to θ.

dx dθ =a d( cosθ+θsinθ ) dθ dx dθ =a( sinθ+sinθ+θcosθ ) dx dθ =a( θcosθ )

We know that,

dy dx = dy dθ dx dθ

Substitute the value of dy dθ and dx dθ .

dy dx = a( θsinθ ) a( θcosθ ) dy dx = sinθ cosθ dy dx =tanθ

Thus, the solution is dy dx =tanθ.


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