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The equation of normal at any point θ to the curve x=acosθ+a θsinθ, y=asinθa θcosθ is always at a distance of

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a
2a units from origin
b
a units from origin
c
12a units from origin
d
5a units from origin

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detailed solution

Correct option is B

x=acosθ+aθsinθ,y=asinθaθcosθdydx=aθsinθaθcosθ=sinθcosθ

Slope of normal =cosθsinθ

Equation of normal at θ

y[asinθaθcosθ]=cosθsinθ[x(acosθ+aθsinθ)]ysinθasin2θ+aθsinθcosθ=xcosθ+acos2θ+aθsinθcosθxcosθ+ysinθa=0

The distance from origin to the normal

=|a|cos2θ+sin2θ=a

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