Find ds/d? for the curves : r2 = a2cos2? etc.,

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Tangent and Normal (II)
(Differential Calculus)

Written by :- Thokchom Sarojkumar Sinha

Find ds/dθ for the curves : (a) r2 = a2cos2θ (b) rn = ancosnθ

Soluion :- (a) r2 = a2cos2θ

We know that ds/dθ = √[r2 + (dr/dθ)2] -------------------------(1)
Differentiating the equation r2 = a2cos2θ with respect to 'θ' , we get
2r dr/dθ = a2 (- sin 2θ).2 = - 2a2sin2θ
or, dr/dθ = - a2sin2θ/r

Using this value in (1) , we get
ds/dθ = √[r2 + (- a2sin2θ/r)2] = √[r4 + a4sin22θ]/r
= ...