---
title: "Ellipse curvature at end of major axis"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:curvature-ellipse-extremity-major-axis"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-ellipse-extremity-major-axis
site: BCT Engineering
---

# Ellipse curvature at end of major axis

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2068 Shrawan
- Video: [RADIUS OF CURVATURE AT EXTREMITY OF MAJOR AXIS IS EQUAL TO HALF OF LATUS RECTUM //EASY EXPALANATION](https://www.youtube.com/watch?v=3amdVc3k1II) — Engineering Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-ellipse-extremity-major-axis
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Show that for the ellipse x²/a² + y²/b² = 1, the radius of curvature at the extremity of the major axis is equal to half of the latus rectum.

## Short answer

At (a,0) (t=0), parametric form gives ρ = b²/a. The latus rectum is 2b²/a, so half is b²/a, proving ρ equals half the latus rectum.

## Answer

Ellipse: x²/a² + y²/b² = 1 with a > b, so the major axis lies along the x-axis, with extremity (a, 0). The latus rectum is 2b²/a, so half of it is b²/a.

**Radius of curvature at (a, 0).** Use the parametric form x = a cos t, y = b sin t. The point (a, 0) corresponds to t = 0.

```text
x′ = −a sin t,  y′ = b cos t
x″ = −a cos t,  y″ = −b sin t
ρ = (x′² + y′²)^(3/2) / (x′y″ − y′x″)
  = (a² sin²t + b² cos²t)^(3/2) / (ab sin²t + ab cos²t)
  = (a² sin²t + b² cos²t)^(3/2) / (ab)
```
At t = 0: ρ = (b²)^(3/2)/(ab) = b³/(ab) = b²/a.

Since the latus rectum is l = 2b²/a, we get

**ρ = b²/a = l/2 = half of the latus rectum.** Shown.

## Check yourself

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

1. For the ellipse with a > b, the latus rectum is
   A. 2b^2/a
   B. 2a^2/b
   C. b^2/a
   D. 2b/a
   Answer: A. Half of it, b^2/a, is the quantity to be matched. (Key point: Latus rectum)

2. The point (a, 0) corresponds to which parameter value in x = a cos t, y = b sin t?
   A. t = π/2
   B. t = 0
   C. t = π
   D. t = π/4
   Answer: B. cos 0 = 1, sin 0 = 0 gives (a, 0). (Key point: Parameter value)

3. x'y'' - y'x'' for the ellipse simplifies to
   A. a^2 + b^2
   B. ab cos t
   C. ab
   D. a^2 b^2
   Answer: C. It equals ab sin^2 t + ab cos^2 t = ab. (Key point: Denominator)

4. At t = 0, rho equals
   A. a^2/b
   B. b^3/a
   C. a/b
   D. b^2/a
   Answer: D. (b^2)^(3/2)/(ab) = b^3/(ab) = b^2/a, half the latus rectum. (Key point: Result)

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