---
title: "Rotation angle to remove xy term"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:rotate-axes-3x2-2xy-3y2-remove-xy"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rotate-axes-3x2-2xy-3y2-remove-xy
site: BCT Engineering
---

# Rotation angle to remove xy term

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2074 Chaitra, 2+3 marks
- Video: [Rotation of Axes Example - Eliminate the xy term](https://www.youtube.com/watch?v=wzGMGNhkFy4) — Symplit Math
- 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-3x2-2xy-3y2-remove-xy
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Through what angle should the axes be rotated to reduce the equation 3x² + 2xy + 3y² − √2 x = 0 into one with the xy term missing? Also obtain the transformed equation.

## Short answer

Set tan2θ = 2h/(a−b) = 2/(0) → 2θ=90°, so θ=45°. Substitute x=(X−Y)/√2, y=(X+Y)/√2 to get 4X²+2Y²−X+Y=0.

## Answer

#### Angle of rotation
Rotate the axes through θ: x = X cosθ − Y sinθ, y = X sinθ + Y cosθ. For ax² + 2hxy + by², the XY coefficient becomes

2h cos2θ − (a − b) sin2θ

and it vanishes when **tan 2θ = 2h/(a − b)**.

Here a = 3, b = 3, 2h = 2, so a − b = 0 and tan 2θ = ∞. Thus 2θ = 90°, **θ = 45°**.

#### Transformed equation
With θ = 45°, x = (X − Y)/√2 and y = (X + Y)/√2.

- x² + y² = X² + Y²
- xy = (X² − Y²)/2
- √2 x = X − Y

Then

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

and the equation 3x² + 2xy + 3y² − √2 x = 0 becomes

**4X² + 2Y² − X + Y = 0**

The XY term is gone. Completing the squares gives 4(X − 1/8)² + 2(Y + 1/4)² = 1/16 + 1/8 = 3/16: an ellipse with centre (1/8, −1/4) in the new axes.

## Check yourself

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

1. What is the condition for removing xy?
   A. tan2θ = (a − b)/2h
   B. tanθ = 2h
   C. tan2θ = 2h/(a − b)
   D. cos2θ = a − b
   Answer: C. Set the XY coefficient 2h cos2θ − (a−b) sin2θ to zero. (Key point: Condition)

2. What angle is needed here?
   A. 30°
   B. 45°
   C. 60°
   D. 90°
   Answer: B. a = b so tan2θ = ∞, 2θ = 90°. (Key point: Angle)

3. What does −sqrt2 x become?
   A. −(X − Y), i.e. −X + Y
   B. −X − Y
   C. −2X
   D. −sqrt2 X
   Answer: A. x = (X−Y)/sqrt2 so sqrt2 x = X − Y. (Key point: Linear term)

4. The transformed equation?
   A. 4X^2 + 2Y^2 + X − Y = 0
   B. 3X^2 + 3Y^2 − X = 0
   C. 2X^2 + 4Y^2 − X + Y = 0
   D. 4X^2 + 2Y^2 − X + Y = 0
   Answer: D. Quadratic part 4X^2 + 2Y^2 with linear part −X + Y. (Key point: Result)

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