---
title: "Asymptotes of a²b²+2ab²x+... curve"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-a2b2-polynomial"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-a2b2-polynomial
site: BCT Engineering
---

# Asymptotes of a²b²+2ab²x+... curve

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2068 Chaitra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-a2b2-polynomial
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the asymptotes of the curve a²b² + 2ab²x + b²x² + a²x² + 2ax³ + x⁴ − x²y² = 0.

## Short answer

The equation simplifies to (a+x)²(b²+x²) = x²y². Asymptotes: vertical x = 0, oblique y = x + a and y = −x − a.

## Answer

Curve: a²b² + 2ab²x + b²x² + a²x² + 2ax³ + x⁴ − x²y² = 0.

Group the terms: b²(a² + 2ax + x²) + x²(a² + 2ax + x²) = (a + x)²(b² + x²). So the curve is

(a + x)²(b² + x²) = x²y², degree 4.

**Parallel to the y-axis:** coefficient of y² is −x²; x² = 0 gives **x = 0**.

**Oblique:**

```text
φ₄(m) = −m² + 1 = 1 − m²
φ₃(m) = 2a
```
Slopes m = ±1. c = −φ₃/φ₄′ = −2a/(−2m) = a/m.

- m = 1: c = a, so y = x + a.
- m = −1: c = −a, so y = −x − a.

**Answer: x = 0, y = x + a, y = −(x + a)** (same as the curve (a+x)²(b²+x²) = x²y²).

## Check yourself

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

1. The curve reduces to:
   A. x^2y^2 = a^2b^2
   B. (a-x)^2 (b^2+x^2) = y^2
   C. (a+x)^2(b^2+x^2) = x^2 y^2
   D. (a+x)(b+x) = xy
   Answer: C. Factor both groups with (a+x)^2. (Key point: Grouping)

2. The vertical asymptote is:
   A. x = a
   B. x = -a
   C. x = 0
   D. y = 0
   Answer: C. Coefficient of y^2 is -x^2. (Key point: Vertical)

3. phi_4 = 1 - m^2 gives slopes:
   A. 2
   B. +-1
   C. 0
   D. 1 twice
   Answer: B. 1 - m^2 = 0. (Key point: Slopes)

4. The asymptotes are:
   A. x = a, y = x
   B. y = x + b
   C. x = 0, y = x + a, y = -x - a
   D. y = a
   Answer: C. c = a/m. (Key point: Answer)

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