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Question

The normal of the curve x=a(cosθ+θsinθ);y=a(sinθθcosθ) at any θ is such that 
  1. It makes a constant angle with x-axis
  2. It passes through the origin
  3. It is at a constant distance from the origin
  4. None of these


Solution

The correct option is C It is at a constant distance from the origin
y=a(sinθθcosθ),x=a(cosθ+θsinθ)
dydθ=a[cosθcosθ+θsinθ]=aθsinθdxdθ=a(sinθ+sinθ+θcosθ)=aθcosθdydx=dydθdxdθ=aθsinθaθcosθ=tanθ
Slope of the tangent = tanθ
Slope of the normal =cotθ
Hence, equation of normal
[yasinθ+aθcosθ]=cosθsinθ[xacosθaθsinθ]ysinθasin2θ+aθsinθcosθ=xcosθ+acos2θ+aθsinθcosθxcosθ+ysinθ=a(sin2θ+cos2θ)xcosθ+ysinθ=a
 Distance from origin =asin2θ+cos2θ=a= constant
 

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