Question

The pedal Equation of the polar curve r^n=a^n cos⁡nθ is given by ______

(a) r^n=pa^n

(b) r^n-1=pa^n

(c) r^n+1=pa^n+1

(d) r^n+1=pa^n

The question was posed to me in an interview.

My question comes from Polar Curves topic in portion Differential Calculus of Engineering Mathematics

Answer 1

Correct answer is (d) r^n+1=pa^n

To elaborate: Taking logarithm for the given curve we get

n log⁡r = n log⁡a + log⁡(cos⁡nθ)

differentiating w.r.t θ, we get

\(\frac{n}{r} \frac{dr}{dθ} = \frac{-n sin⁡nθ}{cos⁡nθ} \rightarrow \frac{1}{r} \frac{dr}{dθ} = -tan⁡θ\)

thus \(cot⁡∅ = cot⁡(\frac{π}{2} + nθ)\rightarrow ∅ = \frac{π}{2} + nθ……(1)\)

from the eqn w.k.t p=r sin ∅

substituting from (1)

p = r sin (\(\frac{π}{2}\) + nθ) = r cos (nθ), but we have r^n = a^n cos⁡nθ

hence dividing them we get \(\frac{p}{r^n} = \frac{r cos (nθ)}{a^n cos⁡nθ}\)

r^n+1=pa^n.