---
title: "Radius of curvature of r^m = a^m cos mθ"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:curvature-polar-rm-cos-mtheta"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-polar-rm-cos-mtheta
site: BCT Engineering
---

# Radius of curvature of r^m = a^m cos mθ

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2070 Asar; 2067 Asoj (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-polar-rm-cos-mtheta
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Show that the radius of curvature at any point (r,θ) of the curve r^m = a^m cos mθ is a^m/((m+1) r^(m−1)).

## Short answer

From the pedal equation p = r^(m+1)/a^m we get dp/dr = (m+1)r^m/a^m. Hence radius of curvature ρ = r dr/dp = a^m / ((m+1) r^(m-1)).

## Answer

Use the pedal form ρ = r dr/dp.

**Pedal equation.** From r^m = a^m cos mθ: (1/r)(dr/dθ) = −tan mθ, so tan φ = r dθ/dr = −cot mθ = tan(π/2 + mθ), giving φ = π/2 + mθ.

p = r sin φ = r cos mθ = r · r^m / a^m = r^(m+1)/a^m.

**Differentiate** with respect to r:

dp/dr = (m+1) r^m / a^m

**Radius of curvature:**

ρ = r (dr/dp) = r · a^m / ((m+1) r^m)

**ρ = a^m / ((m+1) r^(m−1)).** Hence shown.

(Special cases: m = 1 gives the circle r = a cos θ with ρ = a/2; m = 2 gives the lemniscate with ρ = a²/(3r).)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which pedal formula is used for radius of curvature?
   A. rho = r dp/dr
   B. rho = r dr/dp
   C. rho = p dr/dr
   D. rho = dp/dr
   Answer: B. rho = r dr/dp. (Key point: Formula)

2. The pedal equation here is:
   A. p = r^m/a^m
   B. p = a^m r
   C. p = r^(m-1)
   D. p = r^(m+1)/a^m
   Answer: D. From p = r cos m theta. (Key point: Pedal equation)

3. dp/dr =
   A. (m+1) r^m/a^m
   B. r^(m+1)
   C. (m+1) r^(m+1)/a^m
   D. m r^m/a^m
   Answer: A. Differentiate r^(m+1)/a^m. (Key point: Derivative)

4. For m = 1 (circle r = a cos theta), rho =
   A. 2a
   B. a/2
   C. a
   D. a/4
   Answer: B. a/(2 * 1) = a/2. (Key point: Check)

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