---
title: "Pedal equation of r² = a²cos2θ"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:pedal-equation-lemniscate"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/pedal-equation-lemniscate
site: BCT Engineering
---

# Pedal equation of r² = a²cos2θ

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Kartik; 2071 Chaitra (asked 2 times)
- Video: [Pedal Equation of the Curve (Examples 1)  Polar Curves | Engineering Mathematics](https://www.youtube.com/watch?v=b3Ps0_ogtnQ) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/pedal-equation-lemniscate
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the pedal equation of the curve r² = a²cos2θ.

## Short answer

The pedal equation is p a² = r³. Since φ = π/2 + 2θ, p = r cos 2θ and cos 2θ = r²/a² yields p = r³/a².

## Answer

Curve: r² = a² cos 2θ. The pedal equation uses p = r sin φ with tan φ = r dθ/dr.

Take logs: 2 ln r = ln a² + ln cos 2θ. Differentiate with respect to θ:

(2/r)(dr/dθ) = −2 tan 2θ, so r dθ/dr = −cot 2θ = tan(π/2 + 2θ)

Hence φ = π/2 + 2θ, and

p = r sin φ = r cos 2θ.

Since cos 2θ = r²/a²:

p = r³/a²

**Pedal equation: p a² = r³.**

## Check yourself

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

1. For r^2 = a^2 cos 2 theta, phi =
   A. pi/2 + 2 theta
   B. pi - 2 theta
   C. pi/2 - 2 theta
   D. 2 theta
   Answer: A. r d theta/dr = -cot 2 theta. (Key point: Angle)

2. p = r sin phi =
   A. r tan 2 theta
   B. r cos 2 theta
   C. r
   D. r sin 2 theta
   Answer: B. sin(pi/2 + 2 theta) = cos 2 theta. (Key point: Perpendicular)

3. Using cos 2 theta = r^2/a^2:
   A. p = a^2/r
   B. p = r/a^2
   C. p = r^2/a
   D. p = r^3/a^2
   Answer: D. p = r * r^2/a^2. (Key point: Substitution)

## Close variants set in other exams

- [Pedal equation of x^(2/3)+y^(2/3)=a^(2/3)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Apedal-equation-astroid) — Mathematics I, chapter 1, asked 3×

## Related in Mathematics I

- [Pedal equation of r^m = a^m cos mθ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Apedal-equation-rm-cos-m-theta) — Mathematics I, chapter 1, asked 2×
- [Pedal equation of an ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Apedal-equation-ellipse) — Mathematics I, chapter 1
- [Radius of curvature of a² = r²cos2θ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-lemniscate-a2-r2cos2theta) — Mathematics I, chapter 1, asked 2×

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