Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.

Question

Philoid · Accepted 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.