---
title: "Pedal equation of x^(2/3)+y^(2/3)=a^(2/3)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:pedal-equation-astroid"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/pedal-equation-astroid
site: BCT Engineering
---

# Pedal equation of x^(2/3)+y^(2/3)=a^(2/3)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2070 Chaitra
- 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-astroid
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the pedal equation of the curve x^(2/3) + y^(2/3) = a^(2/3).

## Short answer

Parametrize x = a cos³t, y = a sin³t; get p = a sin t cos t, r² = a² – 3p², so the pedal equation is r² = a² – 3p².

## Answer

Astroid: x^(2/3) + y^(2/3) = a^(2/3). Use the parametric form x = a cos³t, y = a sin³t.

dx/dt = −3a cos²t sin t, dy/dt = 3a sin²t cos t, so dy/dx = −tan t.

**Tangent:** y − a sin³t = −tan t (x − a cos³t). Multiply by cos t:

x sin t + y cos t = a sin t cos t (sin²t + cos²t) = a sin t cos t

**Perpendicular from the origin:**

p = a sin t cos t / √(sin²t + cos²t) = a sin t cos t …(1)

**Radius vector:**

r² = x² + y² = a²(cos⁶t + sin⁶t) = a²(1 − 3 sin²t cos²t)

since cos⁶t + sin⁶t = (cos²t + sin²t)³ − 3 sin²t cos²t(cos²t + sin²t). Using (1):

r² = a² − 3p²

**Pedal equation: r² = a² − 3p²** (or 3p² = a² − r²).

## Check yourself

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

1. The astroid is parametrised as:
   A. x = a cos 3t
   B. x = a cos^3 t, y = a sin^3 t
   C. x = a t^3
   D. x = a cos t, y = a sin t
   Answer: B. Satisfies the equation. (Key point: Parametrisation)

2. dy/dx =
   A. -tan t
   B. 1
   C. -cot t
   D. tan t
   Answer: A. (3a sin^2 cos)/(-3a cos^2 sin). (Key point: Slope)

3. p =
   A. a
   B. a sin t cos t
   C. a sin t
   D. a cos t
   Answer: B. Tangent x sin t + y cos t = a sin t cos t. (Key point: p)

4. The pedal equation is:
   A. r^2 = a^2 + 3p^2
   B. r = 3p
   C. p = a r
   D. r^2 = a^2 - 3p^2
   Answer: D. r^2 = a^2(1 - 3 sin^2 cos^2). (Key point: Result)

## Close variants set in other exams

- [Pedal equation of r² = a²cos2θ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Apedal-equation-lemniscate) — 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

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