---
title: "Asymptotes of (x²−y²)²−2(x²+y²)+x−1=0"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-x2-y2-squared-minus"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-x2-y2-squared-minus
site: BCT Engineering
---

# Asymptotes of (x²−y²)²−2(x²+y²)+x−1=0

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2070 Asar; 2069 Chaitra (asked 2 times)
- Video: [Find the asymptotes of the curve x^2(x-y)^2-a^2(x^2+y^2)=0](https://www.youtube.com/watch?v=EUaRiw5AlAk) — Master of Mathematics 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-x2-y2-squared-minus
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the asymptotes of the curve (x²−y²)² − 2(x²+y²) + x − 1 = 0.

## Short answer

The asymptotes are y = x + 1, y = x - 1, y = -x + 1, y = -x - 1. Double roots m=±1 from φ₄(m)=0 lead to equation 4c²-4=0, giving c=±1 for each slope.

## Answer

Curve: (x² − y²)² − 2(x² + y²) + x − 1 = 0. Degree 4.

```text
φ₄(m) = (1 − m²)²
φ₃(m) = 0
φ₂(m) = −2(1 + m²)
φ₁(m) = 1
```
φ₄(m) = 0 gives m = 1 (twice) and m = −1 (twice). Since φ₃ = 0, for a double root use

c² φ₄″(m)/2 + c φ₃′(m) + φ₂(m) = 0.

φ₄ = (1 − m)²(1 + m)², so at m = 1: φ₄″/2 = (1 + m)² = 4. At m = −1: φ₄″/2 = (1 − m)² = 4.

**m = 1:** 4c² + 0 − 4 = 0, so c = ±1. Asymptotes: **y = x + 1, y = x − 1**.

**m = −1:** 4c² + 0 − 4 = 0, so c = ±1. Asymptotes: **y = −x + 1, y = −x − 1**.

**Answer: y = x ± 1 and y = −x ± 1** (four asymptotes).

## Check yourself

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

1. The curve (x^2-y^2)^2 - 2(x^2+y^2) + x - 1 = 0 has degree:
   A. 1
   B. 3
   C. 4
   D. 2
   Answer: C. The first term is degree 4. (Key point: Degree)

2. phi_4(m) = (1-m^2)^2 gives:
   A. m = 1 and -1 once each
   B. m = 2, -2
   C. m = 0 four times
   D. m = 1 twice and m = -1 twice
   Answer: D. Each factor is squared. (Key point: Slopes)

3. Since phi_3 = 0, the double-root equation reduces to:
   A. c phi_4' + phi_2 = 0
   B. c^2 + phi_2 = 0
   C. phi_4 = c
   D. c^2 phi_4''/2 + phi_2 = 0
   Answer: D. The c phi_3' term vanishes. (Key point: Formula)

4. At m = 1, 4c^2 - 4 = 0 gives:
   A. c = 4
   B. c = +-1, giving y = x +- 1
   C. c = +-2
   D. c = 0
   Answer: B. c^2 = 1. (Key point: Result)

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