Question

If φ be the angle which the radius vector of the curve r = f(θ) makes with the tangent prove that  r/p = sinφ(1 + dφ/dθ), where ρ is the radius of curvature of the curve. Also apply the result to show that ρ = a/2 for the circle r = a cosθ

Answer 1

See the geometry given in fig.

In the ∆OPT, ψ = θ + φ                                                                                    ......(1)

ψ is the angle which the tangent to the curve makes at with the initial axis 

θ is the angle which the radius vector OP makes with the initial axis; 

φ is the angle which the radius vector encloses with the tangent at P(r, θ).

Equation (1) implies 

dψ/ds = dθ/ds + dφ/ds

i.e.  

                                                                                .....(2)

In the ∆PNQ, for the limiting arc when Q approaches to P(back), i.e. when δθ → 0

                                                                          

                                                     .....(3)

                                                                                .....(4)

Now on using (4), (2) becomes

                                                                    .....(5)

Further, the circle r = a cosθ implies r₁ = –a sinθ