---
title: "Angle to remove xy term from ax²+2hxy+by²=0"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:rotate-axes-ax2-2hxy-by2"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rotate-axes-ax2-2hxy-by2
site: BCT Engineering
---

# Angle to remove xy term from ax²+2hxy+by²=0

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2068 Shrawan
- Video: [Angle of rotation to remove xy term from the expression ax^2+2hxy+by^2=c](https://www.youtube.com/watch?v=cPdb1-GoriA) — MATHS A TO Z
- 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-ax2-2hxy-by2
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Find the angle through which the axes must be turned so that the equation ax² + 2hxy + by² = 0 may become an equation having no term involving xy.

## Short answer

Rotate by θ such that tan2θ=2h/(a−b). If a=b, θ=45°. The resulting equation is λ₁X²+λ₂Y²=0 with λ₁,λ₂ = ½[(a+b)±√((a−b)²+4h²)].

## Answer

Rotate the axes through an angle θ:

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

#### Substitute into ax² + 2hxy + by² = 0
- x² = X² cos²θ − 2XY sinθ cosθ + Y² sin²θ
- y² = X² sin²θ + 2XY sinθ cosθ + Y² cos²θ
- xy = X² sinθ cosθ + XY (cos²θ − sin²θ) − Y² sinθ cosθ

Collecting the coefficient of XY:

−2a sinθ cosθ + 2h (cos²θ − sin²θ) + 2b sinθ cosθ = 2h cos2θ − (a − b) sin2θ

#### Condition for no XY term
2h cos2θ − (a − b) sin2θ = 0

**tan 2θ = 2h/(a − b)**, so θ = ½ tan⁻¹[2h/(a − b)]

If a = b, then cos2θ = 0 and θ = 45°.

#### Resulting equation
The new coefficients of X² and Y² are

λ₁, λ₂ = ½[(a + b) ± √((a − b)² + 4h²)]

and the equation becomes λ₁X² + λ₂Y² = 0 (the two axes are the bisectors of the angles between the pair of lines).

## Check yourself

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

1. After rotation, the XY coefficient is?
   A. 2h cos2θ − (a − b) sin2θ
   B. 2h sin2θ − (a − b) cos2θ
   C. (a + b) cos2θ
   D. 2h cos2θ + (a − b) sin2θ
   Answer: A. Collect XY terms from x^2, y^2 and xy. (Key point: XY coefficient)

2. What angle removes it?
   A. tanθ = 2h/(a − b)
   B. tan2θ = (a − b)/2h
   C. θ = 90°
   D. tan2θ = 2h/(a − b)
   Answer: D. Set the coefficient to zero. (Key point: Angle)

3. If a = b, what angle?
   A. 0°
   B. 90°
   C. 45°
   D. 30°
   Answer: C. cos2θ = 0 so θ = 45°. (Key point: Special case)

4. The new X^2, Y^2 coefficients are?
   A. a and b
   B. ½[(a+b) ± sqrt((a−b)^2 + 4h^2)]
   C. a + b and a − b
   D. h and −h
   Answer: B. The principal values; their sum is a + b and product ab − h^2. (Key point: New coefficients)

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