---
title: "Product of semi-axes of 5x²+6xy+5y²+12x+4y−4=0"
subject: "Mathematics I"
chapter: 3
key: "semester_1/mathematics_i:conic-5x2-6xy-5y2-product-semi-axes"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/conic-5x2-6xy-5y2-product-semi-axes
site: BCT Engineering
---

# Product of semi-axes of 5x²+6xy+5y²+12x+4y−4=0

Mathematics I · Chapter 3: Plane Analytic Geometry

- Asked in: 2069 Asar
- 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-5x2-6xy-5y2-product-semi-axes
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3

## Question

Prove that the product of the semi-axes of the conic 5x² + 6xy + 5y² + 12x + 4y − 4 = 0 is 3.

## Short answer

Centre (−3/2, 1/2). After rotation the equation becomes 8u²+2v²=12, semi-axes √(3/2) and √6; their product is 3.

## Answer

Here a = 5, h = 3, b = 5, so h² − ab = 9 − 25 < 0: an ellipse.

#### Centre
5x + 3y + 6 = 0 and 3x + 5y + 2 = 0. Solving: x = −3/2, y = 1/2. **Centre: (−3/2, 1/2).**

#### Shift the origin
Constant: g·x₀ + f·y₀ + c = 6(−3/2) + 2(1/2) − 4 = −9 + 1 − 4 = −12, so

5X² + 6XY + 5Y² = 12

#### Rotate
Since a = b, rotate through 45°: X = (u − v)/√2, Y = (u + v)/√2. Then X² + Y² = u² + v² and XY = (u² − v²)/2:

5(u² + v²) + 3(u² − v²) = 12
8u² + 2v² = 12

u²/(3/2) + v²/6 = 1

#### Semi-axes
A = √(3/2), B = √6.

**Product of the semi-axes** = √(3/2 · 6) = √9 = **3**. Hence proved.

(Check with the general result: product = |constant| / √(ab − h²) = 12/√16 = 3.)

## Check yourself

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

1. What is the centre of the conic?
   A. (3/2, −1/2)
   B. (−1/2, 3/2)
   C. (−3/2, 1/2)
   D. (0, 0)
   Answer: C. 5x + 3y + 6 = 0 and 3x + 5y + 2 = 0 give x = −3/2, y = 1/2. (Key point: Centre)

2. Why rotate through 45°?
   A. h = 0
   B. a = b so tan2θ = 2h/(a − b) is infinite
   C. The centre is at the origin
   D. Because 5 + 5 = 10
   Answer: B. When a = b, 2θ = 90°. (Key point: Rotation angle)

3. After rotation what is the equation?
   A. 8u^2 + 2v^2 = 12
   B. 5u^2 + 5v^2 = 12
   C. 8u^2 − 2v^2 = 12
   D. 2u^2 + 8v^2 = 6
   Answer: A. 5(u^2+v^2) + 3(u^2 − v^2) = 8u^2 + 2v^2 = 12. (Key point: Reduced equation)

4. What is the product of semi-axes?
   A. 9
   B. sqrt3
   C. 6
   D. 3
   Answer: D. sqrt((3/2)·6) = sqrt9 = 3, equal to 12/sqrt(ab − h^2) = 12/4. (Key point: Product)

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