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Question

The normal of the curve x=acosθ+θsinθ, y=asinθ-θcosθ at any θ is such that


A

It makes a constant angle with the x-axis

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B

It passes through the origin

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C

It is at a constant distance from the origin

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D

None of the above

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Solution

The correct option is C

It is at a constant distance from the origin


Find the normal of the curve based on given information

Given, x=acosθ+θsinθ , y=asinθ-θcosθ

dydθ=acosθ-cosθ+θsinθ=aθsinθ

dxdθ=a-sinθ+sinθ+θcosθ=aθcosθ

dydx=aθsinθaθcosθ=tanθ

Hence, the slope of the normal is -cotθ.

The equation of the normal is given as

y-asinθ-θcosθ=-cosθsinθx-acosθ+θsinθ

ysinθ-asin2θ-aθsinθcosθ=-xcosθ+acos2θ-aθcosθsinθ

xcosθ+ysinθ-asin2θ+cos2θ=0

xcosθ+ysinθ-a=0

The distance of the origin to the normal is given as

ON=0+0-acos2θ+sin2θ

ON=a

So, the distance of the origin from the normal to the curve is constant.

Hence, option C is the correct answer.


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