---
title: "Product of semi-axes of x²−4xy+5y²=2"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:conic-x2-4xy-5y2-product-semi-axes"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/conic-x2-4xy-5y2-product-semi-axes
site: BCT Engineering
---

# Product of semi-axes of x²−4xy+5y²=2

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2074 Chaitra, 5 marks
- Video: [Product of semi-axes, conic](https://www.youtube.com/watch?v=NTxjK-Rr5Uk) — Dr Pokhrel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/conic-x2-4xy-5y2-product-semi-axes
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Find the product of semi-axes of the conic x² − 4xy + 5y² = 2.

## Short answer

The conic is an ellipse centred at the origin. Its matrix eigenvalues satisfy λ²−6λ+1=0, product λ₁λ₂=1. The product of the semi-axes is √(4/(λ₁λ₂))=2.

## Answer

Here a = 1, h = −2, b = 5, and the centre is the origin (no linear terms), so

x² − 4xy + 5y² = 2

h² − ab = 4 − 5 = −1 < 0, so the conic is an ellipse.

#### Rotate to remove the xy term
After rotating through the principal angle (tan 2θ = 2h/(a − b) = 1), the equation becomes λ₁u² + λ₂v² = 2, where λ₁, λ₂ are the roots of

λ² − (a + b)λ + (ab − h²) = 0
λ² − 6λ + 1 = 0

so λ₁ + λ₂ = 6 and λ₁λ₂ = 1. (The roots are 3 ± 2√2.)

#### Semi-axes
u²/(2/λ₁) + v²/(2/λ₂) = 1

Semi-axes: A = √(2/λ₁), B = √(2/λ₂).

**Product of semi-axes** = AB = √(4/(λ₁λ₂)) = 2/√1 = **2**

Individually, A = √(2/(3 − 2√2)) = √2(√2 + 1) = 2 + √2 and B = √2(√2 − 1) = 2 − √2, whose product is 4 − 2 = 2. This agrees.

## Check yourself

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

1. For x^2 − 4xy + 5y^2 = 2, what is the conic?
   A. A hyperbola
   B. A parabola
   C. An ellipse centred at the origin, since h^2 − ab = −1 < 0
   D. A pair of lines
   Answer: C. No linear terms so centre is origin; h^2 − ab = 4 − 5 < 0. (Key point: Type)

2. What equation gives the coefficients after rotation?
   A. λ^2 − 6λ − 1 = 0
   B. λ^2 − (a+b)λ + (ab − h^2) = 0, i.e. λ^2 − 6λ + 1 = 0
   C. λ^2 + 6λ + 1 = 0
   D. λ^2 − 5λ + 4 = 0
   Answer: B. Sum a + b = 6, product ab − h^2 = 1. (Key point: Principal coefficients)

3. In λ₁u^2 + λ₂v^2 = 2, what are the semi-axes?
   A. sqrt(2/λ₁) and sqrt(2/λ₂)
   B. 2/λ₁ and 2/λ₂
   C. sqrt(λ₁/2) and sqrt(λ₂/2)
   D. λ₁ and λ₂
   Answer: A. Writing as u^2/(2/λ₁) + v^2/(2/λ₂) = 1. (Key point: Semi-axes)

4. What is the product of the semi-axes?
   A. 1
   B. sqrt2
   C. 4
   D. 2
   Answer: D. AB = sqrt(4/(λ₁λ₂)) = sqrt(4/1) = 2. (Key point: Product)

## Related in Mathematics I

- [Product of semi-axes of 5x²+6xy+5y²+12x+4y−4=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-5x2-6xy-5y2-product-semi-axes) — Mathematics I, chapter 3
- [Axes of conic 14x²−4xy+11y²−44x−58y+71=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-14x2-4xy-11y2-axes) — Mathematics I, chapter 3, asked 2×
- [Centre, axes of 9x²+4xy+6y²−22x−16y+9=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-9x2-4xy-6y2-centre-axes) — Mathematics I, chapter 3, asked 7×
- [Equation and length of axes of x²−4xy−2y²+10x+4y=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-x2-4xy-2y2-equation-axes) — Mathematics I, chapter 3
- [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×
- [Axes along bisectors of 7x²+4xy+4y²=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Atransform-pair-of-lines-7x2-4xy-4y2) — Mathematics I, chapter 3