Question

If r=b e^θ cot⁡a, where a, b are constants then ‘s’ is represented by which one of the following equation?

(a) s=r sec(a)+c

(b) s=r cos(a)+c

(c) s=r+c

(d) s=r tan(a)+c

I got this question in a job interview.

My doubt stems from Derivative of Arc Length topic in division Differential Calculus of Engineering Mathematics

Answer 1

Right answer is (a) s=r sec(a)+c

To explain I would say: r=b e^θ cot⁡a

\(\frac{dr}{dθ} = b e^{θ cot⁡a}(cot a)=r cot(a) \)

w.k.t \(\frac{ds}{dr} = \sqrt{1+r^2 (\frac{dθ}{dr})^2} \,where\, \frac{dθ}{dr}=\frac{1}{r} \,tan⁡a\)

\(\frac{ds}{dr} = \sqrt{1+tan^2 a} = sec \,a \rightarrow s = \int sec \,a \,dr \, = r sec(a)+c,\) where c is constant of integration.