Answer :

We have x = a cosθ + a θ sinθ,


And y = a sinθ – aθ cosθ



So,


Then, Slope of the normal at any point θ is .


The equation of the normal at a given point (x,y) is:


y - a sinθ + aθ cosθ = (x - a cosθ - a θ sinθ)


ysinθ asin2θ + aθ sinθ cosθ = - x cosθ + acos2θ + aθ sinθ cosθ


xcosθ + ysinθ a(sin2θ + cos2θ ) = 0


xcosθ + ysinθ a = 0


Now, the perpendicular distance of the normal from the origin is


, which is independent of θ .


Therefore, the perpendicular distance of the normal from the origin is constant.


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