---
title: "Rotate axes 30° in x²+2√3xy−y²=2a²"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:rotate-axes-x2-2sqrt3xy-y2-30"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rotate-axes-x2-2sqrt3xy-y2-30
site: BCT Engineering
---

# Rotate axes 30° in x²+2√3xy−y²=2a²

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2073 Chaitra
- Video: [When the axes are rotated through an angle of 30 find the transformed equation of x²+2√3xy-y²=2a²](https://www.youtube.com/watch?v=hqYsQ58YIdY) — AIMSDARETOSUCCESS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/rotate-axes-x2-2sqrt3xy-y2-30
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

What does the equation x² + 2√3xy − y² = 2a² become when the axes are turned through an angle 30° to the original axes?

## Short answer

With θ=30°, substitute x=(√3 X−Y)/2, y=(X+√3 Y)/2. Simplifying gives X²−Y²=a², a rectangular hyperbola.

## Answer

For a rotation of the axes through θ = 30°:

x = X cos30° − Y sin30° = (√3 X − Y)/2
y = X sin30° + Y cos30° = (X + √3 Y)/2

#### Substitute
- x² = (3X² − 2√3 XY + Y²)/4
- y² = (X² + 2√3 XY + 3Y²)/4
- xy = (√3X − Y)(X + √3Y)/4 = (√3X² + 2XY − √3Y²)/4

Then

- x² − y² = (2X² − 4√3 XY − 2Y²)/4
- 2√3 xy = (6X² + 4√3 XY − 6Y²)/4

Adding:

x² + 2√3xy − y² = (8X² − 8Y²)/4 = 2X² − 2Y²

So 2X² − 2Y² = 2a², that is

**X² − Y² = a²**

The xy term has gone: the curve is a rectangular hyperbola with semi-axes a and a.

## Check yourself

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

1. What are x and y?
   A. x = (X − sqrt3 Y)/2, y = (sqrt3 X + Y)/2
   B. x = (X − Y)/sqrt2, y = (X + Y)/sqrt2
   C. x = (sqrt3 X − Y)/2, y = (X + sqrt3 Y)/2
   D. x = (X + sqrt3 Y)/2, y = (sqrt3 X − Y)/2
   Answer: C. x = X cos30° − Y sin30°, y = X sin30° + Y cos30°. (Key point: Formulae for 30°)

2. What happens to XY terms?
   A. They double
   B. They cancel when x^2 − y^2 and 2sqrt3 xy are added
   C. They stay
   D. They become X^2
   Answer: B. −4sqrt3 XY + 4sqrt3 XY = 0. (Key point: Cross terms)

3. What does the equation become?
   A. X^2 − Y^2 = a^2
   B. X^2 + Y^2 = a^2
   C. 2X^2 − 2Y^2 = a^2
   D. X^2 − Y^2 = 2a^2
   Answer: A. Left side = 2X^2 − 2Y^2 = 2a^2. (Key point: Result)

## Close variants set in other exams

- [Rotate axes 45° in 3x²+3y²+2xy=2](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arotate-axes-3x2-3y2-2xy-45) — Mathematics I, chapter 3, asked 5×
- [Rotate axes by tan⁻¹2 in 4xy−3x²−a²=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arotate-axes-4xy-3x2-tan-inverse-2) — Mathematics I, chapter 3, asked 5×

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