---
title: "Asymptotes of a²y²+x²y²−a²x²+2ax³−x⁴=0"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-a2y2-quartic"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-a2y2-quartic
site: BCT Engineering
---

# Asymptotes of a²y²+x²y²−a²x²+2ax³−x⁴=0

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2071 Shrawan, 5 marks
- Video: [ASYMPTOTES | Short Revision Of ASYMPTOTES | Differential Calculus Revision By @gajendrapurohit​](https://www.youtube.com/watch?v=aC3beQFariY) — Dr.Gajendra Purohit - GATE / IIT JAM / CSIR NET
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-a2y2-quartic
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the asymptotes of the curve a²y² + x²y² − a²x² + 2ax³ − x⁴ = 0.

## Short answer

Set highest-degree terms: slopes m = ±1. Then c = a/m gives asymptotes y = x − a and y = −x + a.

## Answer

Curve: a²y² + x²y² − a²x² + 2ax³ − x⁴ = 0. Degree 4.

**Parallel to the axes:** the coefficient of y² is a² + x², which never vanishes for real x, so no asymptote parallel to the y-axis. The highest power of x is x⁴ with a constant coefficient (−1), so no asymptote parallel to the x-axis.

**Oblique asymptotes:**

```text
φ₄(m) = m² − 1
φ₃(m) = 2a
φ₂(m) = a²m² − a²
```
φ₄(m) = 0 gives m = 1 and m = −1 (both simple). With c = −φ₃(m)/φ₄′(m) and φ₄′ = 2m:

- m = 1: c = −2a/2 = −a. Asymptote: y = x − a.
- m = −1: c = −2a/(−2) = a. Asymptote: y = −x + a.

(Check: the curve is y² = x²(x − a)²/(x² + a²), so y = ±x(x − a)/√(x² + a²) ≈ ±(x − a) for large x.)

**Answer: y = x − a and y = −(x − a), i.e. x − y = a and x + y = a.**

## Check yourself

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

1. Why no asymptote parallel to the y-axis?
   A. The coefficient of y^2, a^2 + x^2, never vanishes for real x
   B. a is a constant
   C. y^2 absent
   D. degree is 4
   Answer: A. a^2 + x^2 > 0. (Key point: Axis-parallel)

2. phi_4(m) = m^2 - 1 gives:
   A. m = 2
   B. m = 0
   C. m = 1 and m = -1
   D. m = 1 twice
   Answer: C. Two simple roots. (Key point: Slopes)

3. With c = -phi_3(m)/phi_4'(m), phi_3 = 2a, phi_4' = 2m, for m = 1 c =
   A. 2a
   B. a
   C. -a
   D. -2a
   Answer: C. -2a/2. (Key point: Intercepts)

4. The asymptotes are:
   A. y = a
   B. y = x - a and y = -x + a
   C. y = x + a and y = -x - a
   D. x = a
   Answer: B. c = -a for m = 1 and c = a for m = -1. (Key point: Answer)

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