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Question

Find the slope of the normal to the curve x=acos3θ,y=asin3θ at θ=π4.

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Solution

It is given that x=acos3θ,y=asin3θ

dxdθ=3acos2θ(sinθ)=3acos2θsinθ

& dydθ=3asin2θ(cosθ)

dydx=(dydθ)(dxdθ)=3asin2θcosθ3acos2θsinθ=sinθcosθ=tanθ

Therefore, the slope of the tangent at θ=π4 is given by,

(dydx)θ=π/4=(tanθ)θ=π/4=1

Hence, the slope of the normal at θ=π4 is ,

=1slope of the tangent atθ=π4=11=1

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