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Question

The normal to the curve x=acosθ+sinθ,y=asinθ-θcosθ at any point θ is such that


A

It is a constant distance from the origin

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B

It passes through aπ2,-a

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C

It makes π2-θ with the x-axis

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D

It passes through the origin

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Solution

The correct option is A

It is a constant distance from the origin


Find the normal to the curve

We know that equation of normal is

y-y0=-1dydxx-x0

dydx=tanθ

The equation of the normal at θ is

y-asinθ-θcosθ=-1tanθx-acosθ+sinθysinθ-asin2θ+asinθcosθ=-xcosθ+acos2θ+asinθcosθysinθ+xcosθ-asin2θ+cos2θ=0ysinθ+xcosθ-a=0

It is an equation of straight line which is at a constant distance from the origin 0,0

Hence, the correct option is A.


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