---
title: "Asymptotes of x(x−y)²−3(x²−y²)+8y=0"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-x-x-minus-y-squared"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-x-x-minus-y-squared
site: BCT Engineering
---

# Asymptotes of x(x−y)²−3(x²−y²)+8y=0

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Chaitra; 2070 Chaitra; 2069 Asar (asked 3 times)
- 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-x-x-minus-y-squared
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the asymptotes of the curve x(x−y)² − 3(x²−y²) + 8y = 0.

## Short answer

Coefficient of y² is x+3, giving vertical asymptote x = −3. φ₃(m) = (1−m)² = 0 yields double root m=1. Solving c² + 6c + 8 = 0 gives c = −2, −4, so asymptotes y = x−2, y = x−4.

## Answer

Curve: x(x − y)² − 3(x² − y²) + 8y = 0. Expanding gives x³ − 2x²y + xy² − 3x² + 3y² + 8y = 0, degree 3.

```text
φ₃(m) = (1 − m)²
φ₂(m) = −3(1 − m²) = 3m² − 3
φ₁(m) = 8m
```
**Parallel to y-axis:** the coefficient of the highest power of y (y²) is x + 3. Putting it to zero gives the asymptote **x = −3**. (φ₃ has degree 2 in m, which agrees with one asymptote parallel to the y-axis.)

**Slopes:** φ₃(m) = (1 − m)² = 0 gives m = 1 twice.

For the double root:
φ₃″(m)/2 = 1, φ₂′(1) = 6m = 6, φ₁(1) = 8.

c² + 6c + 8 = 0, so (c + 2)(c + 4) = 0, c = −2 or −4.

Asymptotes: **y = x − 2** and **y = x − 4**.

**Answer: x + 3 = 0, y = x − 2, y = x − 4.**

## Check yourself

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

1. How is the asymptote parallel to the y-axis found for x(x-y)^2 - 3(x^2-y^2) + 8y = 0?
   A. Solve phi_3(m) = 0
   B. Equate the coefficient of the highest power of y (y^2, which is x + 3) to 0
   C. Equate the coefficient of the highest power of x to 0
   D. Put y = 0
   Answer: B. The coefficient of y^2 is x + 3, giving x = -3. (Key point: Parallel to y-axis)

2. phi_3(m) = (1-m)^2 gives which slope?
   A. m = -1 twice
   B. m = 0
   C. m = 1 twice
   D. m = 1 and m = -1
   Answer: C. (1-m)^2 = 0 gives m = 1 repeated. (Key point: Slope)

3. For m = 1, c^2 + 6c + 8 = 0. What are the values of c?
   A. 2 and 4
   B. 2 and -4
   C. -1 and -8
   D. -2 and -4
   Answer: D. (c+2)(c+4) = 0 gives c = -2, -4. (Key point: Intercept equation)

4. Which is the complete set of asymptotes?
   A. y = x - 2, y = x - 4 only
   B. x = -3, y = -x - 2, y = -x - 4
   C. x = 3, y = x + 2, y = x + 4
   D. x = -3, y = x - 2, y = x - 4
   Answer: D. One vertical line and two oblique lines of slope 1. (Key point: Final answer)

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