---
title: "Rotation to change 9x²−2√3xy+7y²=10"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:rotate-axes-9x2-sqrt3xy-7y2"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rotate-axes-9x2-sqrt3xy-7y2
site: BCT Engineering
---

# Rotation to change 9x²−2√3xy+7y²=10

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2067 Asoj
- 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-9x2-sqrt3xy-7y2
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Through what angle should the axes be rotated so that the equation 9x² − 2√3xy + 7y² = 10 may be changed to 3x² + 5y² = 5?

## Short answer

Eigenvalues λ²−16λ+60=0 → λ=6,10. To get 6X²+10Y²=10 (or 3X²+5Y²=5), X-axis must align with λ=6. tan2θ=2h/(a−b)=−√3 → 2θ=120° or −60°. θ=60° gives coefficient 6.

## Answer

We want the angle θ for which 9x² − 2√3 xy + 7y² = 10 becomes 3X² + 5Y² = 5 after rotation.

#### Step 1: principal values
The coefficients of X² and Y² after rotation are the roots of λ² − (a + b)λ + (ab − h²) = 0 with a = 9, b = 7, h = −√3:

λ² − 16λ + (63 − 3) = 0 → λ² − 16λ + 60 = 0 → λ = 6, 10

So the reduced form is 6X² + 10Y² = 10 or 3X² + 5Y² = 5, as required. For this to happen, the X-axis must be along the eigen-direction of λ = 6.

#### Step 2: angle
For a rotation through θ, the coefficient of X² is

a cos²θ + 2h sinθ cosθ + b sin²θ = 9cos²θ − 2√3 sinθ cosθ + 7 sin²θ.

We need this to equal 6. Also the XY term must vanish: tan 2θ = 2h/(a − b) = −2√3/2 = −√3, so 2θ = 120° (or −60°), giving θ = 60° or θ = −30°.

Test θ = 60° (cos = 1/2, sin = √3/2):

9(1/4) − 2√3(√3/4) + 7(3/4) = 9/4 − 6/4 + 21/4 = 24/4 = 6 ✓

(Test θ = −30° gives 10, which would produce 5X² + 3Y² = 5, not the required form.)

**Answer: the axes must be rotated through 60° (anticlockwise).**

#### Verification
With x = (X − √3 Y)/2, y = (√3 X + Y)/2, substitution gives 6X² + 10Y² = 10, i.e. 3X² + 5Y² = 5.

## Check yourself

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

1. Which equation gives the new coefficients?
   A. λ^2 − 16λ + 66 = 0
   B. λ^2 + 16λ + 60 = 0
   C. λ^2 − 60λ + 16 = 0
   D. λ^2 − 16λ + 60 = 0, so λ = 6, 10
   Answer: D. ab − h^2 = 63 − 3 = 60. (Key point: Principal values)

2. Why must X-axis go along λ = 6?
   A. It needs 10 for X^2
   B. Any axis works
   C. 3X^2 + 5Y^2 = 5 needs coefficient 6 for X^2 (6X^2 + 10Y^2 = 10)
   D. 6 is the smaller angle
   Answer: C. Dividing by 2 gives 3 and 5. (Key point: Matching)

3. Which angle satisfies tan2θ = −sqrt3 and gives X^2 coefficient 6?
   A. −30°
   B. 60°
   C. 30°
   D. 120°
   Answer: B. θ = 60° gives 9/4 − 6/4 + 21/4 = 6; −30° gives 10. (Key point: Angle)

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