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

# Pedal equation of an ellipse

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Chaitra, 5 marks
- Video: [pedal equation of elipse differential calculus and its application](https://www.youtube.com/watch?v=_DhmcuCfr6I) — CIVILIAN WORLD
- 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-ellipse
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the pedal equation of the ellipse x²/b² + y²/a² = 1.

## Short answer

With pole at centre, the pedal equation is 1/p² = (a² + b² − r²)/(a²b²), equivalently p² = a²b²/(a² + b² − r²).

## Answer

**Assumption:** the pole is the centre and a > b; so the semi-axes are b (along x) and a (along y). The result for the focus is also given.

Ellipse: x²/b² + y²/a² = 1. The tangent at (x₁, y₁) is x x₁/b² + y y₁/a² = 1.

The perpendicular from the origin to this line is

p = 1 / √(x₁²/b⁴ + y₁²/a⁴), so 1/p² = x₁²/b⁴ + y₁²/a⁴ …(1)

Also r² = x₁² + y₁² and x₁²/b² + y₁²/a² = 1. Put X = x₁²/b². Then y₁² = a²(1 − X) and

r² = b²X + a²(1 − X), so X = (a² − r²)/(a² − b²).

Now (1) becomes

```text
1/p² = X/b² + (1−X)/a²
     = [a²X + b²(1−X)] / (a²b²)
     = [b² + (a²−b²)X] / (a²b²)
     = (a² + b² − r²) / (a²b²)
```
**Pedal equation (pole at centre): 1/p² = (a² + b² − r²)/(a²b²), i.e. p² = a²b²/(a² + b² − r²).**

(With the pole at a focus, the pedal equation is b²/p² = 2a/r − 1.)

## Check yourself

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

1. The perpendicular from the origin to the tangent gives:
   A. p^2 = x1^2 + y1^2
   B. p = x1 + y1
   C. 1/p^2 = x1^2/b^4 + y1^2/a^4
   D. 1/p = x1/b^2
   Answer: C. Distance formula. (Key point: Tangent distance)

2. Putting X = x1^2/b^2 and using r^2 = x1^2 + y1^2, X =
   A. r^2/a^2
   B. (r^2 - b^2)/(a^2-b^2)
   C. 1 - r^2
   D. (a^2 - r^2)/(a^2 - b^2)
   Answer: D. r^2 = b^2 X + a^2(1-X). (Key point: Elimination)

3. The centre pedal equation is:
   A. p^2 = a^2 + b^2
   B. 1/p^2 = (a^2 + b^2 - r^2)/(a^2 b^2)
   C. 1/p^2 = (a^2 - b^2)/r^2
   D. p = r
   Answer: B. Simplification of 1/p^2. (Key point: Result)

4. The focal pedal equation is:
   A. p = r
   B. b^2/p^2 = 2a/r - 1
   C. a^2/p^2 = 2b/r
   D. 1/p = r
   Answer: B. Different pole gives different form. (Key point: Pole choice)

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