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

First, let us find the slope of the tangent to the curve at a point θ.

`m_T=dy/dx=frac(dy/(d theta))(dx/(d theta)`

=-`(asinθ+asinθ+aθcosθ)`/`(acosθ−acosθ+aθsinθ)`

Since, normal is always perpendicular to the tangent, slope of the normal will be

`m_N=-costheta/sintheta`

Normal passes through the point (x,y) given in the question. Using the equation of the line in slope-point form,

⇒`ysinθ−asin2θ+aθsinθcosθ=acos2θ+aθsinθcosθ−xcosθ`

⇒`xcosθ+ysinθ=a(cos2θ+sin2θ)=a`

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