---
title: "Asymptotes of (a+x)²(b²+x²)=x²y²"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-a-plus-x-squared"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-a-plus-x-squared
site: BCT Engineering
---

# Asymptotes of (a+x)²(b²+x²)=x²y²

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Kartik
- 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-a-plus-x-squared
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the asymptotes of the curve (a+x)²(b²+x²) = x²y².

## Short answer

Vertical asymptote x = 0 from coefficient of y². Oblique slopes m = ±1 from highest-degree terms; c = a/m gives y = x + a and y = −x − a.

## Answer

Curve: (a + x)²(b² + x²) = x²y², i.e. x²y² − (a + x)²(b² + x²) = 0. Expanding the second term gives x⁴ + 2ax³ + (a² + b²)x² + 2ab²x + a²b², so the degree is 4.

**Parallel to the y-axis:** the coefficient of the highest power of y (y²) is x². Putting x² = 0 gives the asymptote **x = 0** (counted twice).

**Parallel to the x-axis:** the highest power of x is x⁴ with a constant coefficient, so there is none.

**Oblique asymptotes:**

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

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

(Check: y = ±(a + x)√(b² + x²)/x ≈ ±(x + a)(1 + b²/2x² ...) ≈ ±(x + a).)

**Answer: x = 0, y = x + a, y = −(x + a).**

## Check yourself

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

1. Coefficient of y^2 is x^2, giving:
   A. none
   B. x = a
   C. x = 0 (counted twice)
   D. y = 0
   Answer: C. x^2 = 0. (Key point: Vertical)

2. Why is there no horizontal asymptote?
   A. degree 2
   B. The highest power of x, x^4, has a constant coefficient
   C. a is zero
   D. y missing
   Answer: B. Constant cannot be equated to zero. (Key point: Horizontal)

3. With c = a/m, m = 1 gives:
   A. y = -x + a
   B. y = x - a
   C. y = x
   D. y = x + a
   Answer: D. c = a. (Key point: Slopes and c)

4. m = -1 gives:
   A. y = a
   B. y = -x + a
   C. y = -x - a
   D. y = x - a
   Answer: C. c = -a. (Key point: Other)

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