Q. 53.9( 25 Votes )
Show that the nor
We have x = a cosθ + a θ sinθ,
And y = a sinθ – aθ cosθ
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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