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Question

Find the equations of the tangent and normal at θ=π2 to the curve x=a(θ+sinθ),y=a(1+cosθ)

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Solution

dxdθ=a(1+cosθ)=2acos2θ2
dydθ=asinθ=2asinθ2cosθ2
dydx=dydθdxdθ=tanθ2
slope m=(dydx)θ=π2=tanπ4=1.
Also for θ=π2 The point on the curve is (aπ2+a,a)
Hence, the equation of the tangent at θ=π2 is ya=(1)[xa(π2+1)]
(i.e.,) x+y=12aπ+2a (or) x+y12aπ2a=0
Equation of the normal at this point is
ya=(1)[xa(π2+1)]
or
xy12aπ=0

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