Show that the normal at any point `theta` to the curve `x=acostheta+athetasintheta,\ y=asintheta-a\ thetacostheta` is at a constant distance from the origin.

Explanation

Given curve is :`x = a cos theta + a theta sin, theta, and y = a sin theta - a theta cos theta`
` rArr (dx)/(d theta) =- a sin theta + a (theta cos theta + sin theta)`
` =- a sin theta + a theta cos theta + a sin theta`
` rArr (dx)/(d theta) = a theta cos theta`
and ` (dy)/(d theta) = a cos theta - a [theta(-sin theta)+cos theta]`
`=a cos theta + a theta sin theta - a cos theta`
` rArr (dy)/(d theta) = a theta sin theta`
Slope of tangent at point ` theta`,
` (dy)/(dx) = (dy)/(d theta) xx(d theta)/(dx) = ( a theta sin theta)/(a theta cos theta) = tan theta`
` :." slope of normal at point "theta`,
` = 1/((dy)/(dx))=- 1/(tan theta) =- cot theta =- (cos theta)/(sin theta)`
Equation of normal at point (x, y)
` y - (a sin theta - a theta cos theta) =- (cos theta)/(sin theta)`
` =- (cos theta)/(sin theta) [x - (a cos theta+ a theta sin theta)]`
` rArr y sin theta - a sin^(2) theta + a theta sin theta cos theta`
` =- x cos theta+ a cos^(2) theta + a theta sin theta* cos theta`
`rArr x cos theta + y sin theta = a (sin^(2)theta+ cos^(2)theta)" "(.:' sin^(2)+ cos^(2) theta = 1)`
` x cos theta + y sin theta = a `
` x cos theta+y sin theta - a = 0`
Now, distance of normal from origin
` = (|-a|)/(sqrt(cos^(2) theta + sin^(2) theta)`
` rArr = (|-a|)/sqrt 1=|-a|" "(.:' cos^(2) theta+ sin^(2) theta = 1)`
which is independent to ` theta`. Therefore normal is at constant distance from origin.