---
title: "Rotate axes 45° in 3x²+3y²+2xy=2"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:rotate-axes-3x2-3y2-2xy-45"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rotate-axes-3x2-3y2-2xy-45
site: BCT Engineering
---

# Rotate axes 45° in 3x²+3y²+2xy=2

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2073 Shrawan; 2070 Chaitra (asked 2 times)
- Video: [When d axes r rotated throu an angle Pi/6, find the transformed eq'n of x²+2 _/3 xy - y²=2a²@EAG](https://www.youtube.com/watch?v=2rWbgI88mlw) — Elite Academy Guntur
- 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-3x2-3y2-2xy-45
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

What does the equation 3x² + 3y² + 2xy = 2 become when the axes are turned through an angle of 45° to the original axes?

## Short answer

After rotation through 45°, the equation becomes 2X² + Y² = 1, an ellipse with semi-axes 1/√2 along the new X-axis and 1 along the new Y-axis.

## Answer

For a rotation of the axes through θ, replace

x = X cosθ − Y sinθ, y = X sinθ + Y cosθ

With θ = 45°: x = (X − Y)/√2, y = (X + Y)/√2.

#### Substitution
- x² = (X − Y)²/2, y² = (X + Y)²/2, so x² + y² = X² + Y².
- xy = (X² − Y²)/2.

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

So the equation 3x² + 3y² + 2xy = 2 becomes

4X² + 2Y² = 2, that is **2X² + Y² = 1**

(or X²/(1/2) + Y²/1 = 1). The xy term has disappeared; the curve is an ellipse with semi-axes 1/√2 along the new X-axis and 1 along the new Y-axis.

## Check yourself

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

1. For axes turned through 45°, which substitution is used?
   A. x = (X + Y)/sqrt2, y = (X + Y)/sqrt2
   B. x = (X − Y)/sqrt2, y = (X + Y)/sqrt2
   C. x = X/2 − Y/2, y = X/2 + Y/2
   D. x = X + Y, y = X − Y
   Answer: B. x = X cosθ − Y sinθ and y = X sinθ + Y cosθ with θ = 45°. (Key point: Rotation formulae)

2. What do x^2 + y^2 and xy become?
   A. X^2 + Y^2 and (X^2 − Y^2)/2
   B. X^2 − Y^2 and (X^2 + Y^2)/2
   C. 2X^2 and XY
   D. X^2 + Y^2 and XY
   Answer: A. Rotation preserves x^2 + y^2, and xy = (X^2 − Y^2)/2. (Key point: Useful identities)

3. What does the equation become?
   A. 4X^2 + 2Y^2 = 1
   B. 3X^2 + Y^2 = 2
   C. X^2 + 2Y^2 = 1
   D. 2X^2 + Y^2 = 1
   Answer: D. 3(X^2 + Y^2) + (X^2 − Y^2) = 4X^2 + 2Y^2 = 2, i.e. 2X^2 + Y^2 = 1. (Key point: Result)

## Close variants set in other exams

- [Rotate axes 30° in x²+2√3xy−y²=2a²](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arotate-axes-x2-2sqrt3xy-y2-30) — 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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