---
title: "Asymptotes of (x+y)²(x+2y+2)=x+9y−2"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-x-plus-y-squared"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-x-plus-y-squared
site: BCT Engineering
---

# Asymptotes of (x+y)²(x+2y+2)=x+9y−2

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Chaitra, 5 marks; 2067 Asar; 2067 Asoj (asked 3 times)
- Video: [Find all the asymptotes of (x+y)^2(x+2y+2)=x+9y+2 | asymptotes of (x+y)2(x+2y+2)=x+9y+2](https://www.youtube.com/watch?v=up8VCWNUrDM) — Nitin Kumar Gupta Classes
- 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-plus-y-squared
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the asymptotes of the curve (x+y)²(x+2y+2) = x+9y−2.

## Short answer

Slopes from φ₃(m) = (1+m)²(1+2m) = 0: m = −1/2 (simple) and m = −1 (double). For m=−1/2, c = −1 gives x+2y+2=0. For m=−1, c = ±2√2 gives x+y=±2√2.

## Answer

Curve: (x + y)²(x + 2y + 2) = x + 9y − 2, i.e. (x+y)²(x+2y+2) − x − 9y + 2 = 0. Degree 3. Recall: slopes from φ₃(m) = 0; simple root: c = −φ₂(m)/φ₃′(m); double root: c² φ₃″/2 + c φ₂′ + φ₁ = 0.

```text
φ₃(m) = (1+m)²(1+2m)
φ₂(m) = 2(1+m)²
φ₁(m) = −(1 + 9m)
φ₀    = 2
```
φ₃(m) = 0 gives m = −1 (twice) and m = −1/2 (once).

**m = −1/2 (simple):**
φ₃′(m) = 2(1+m)(1+2m) + 2(1+m)² = 0 + 2(1/4) = 1/2
φ₂(−1/2) = 2(1/4) = 1/2
c = −(1/2)/(1/2) = −1

Asymptote: y = −x/2 − 1, i.e. **x + 2y + 2 = 0**.

**m = −1 (double):**
φ₃″(m)/2 at m = −1 equals (1 + 2m) = −1; φ₂′(−1) = 4(1+m) = 0; φ₁(−1) = −(1 − 9) = 8.

−c² + 0 + 8 = 0, so c = ±2√2.

Asymptotes: **y = −x + 2√2** and **y = −x − 2√2**, i.e. x + y = ±2√2.

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

## Check yourself

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

1. For (x+y)^2(x+2y+2) = x+9y-2, phi_3(m) = (1+m)^2(1+2m). What are the slopes?
   A. m = -1 only
   B. m = -1 (twice), m = -1/2 (once)
   C. m = 1, m = 1/2
   D. m = -1 (once), m = -1/2 (twice)
   Answer: B. (1+m)^2 gives -1 twice and (1+2m) gives -1/2 once. (Key point: Slopes)

2. For the simple root m = -1/2, c = -phi_2(m)/phi_3'(m). What is c?
   A. 1/2
   B. -2
   C. 1
   D. -1
   Answer: D. phi_2(-1/2) = 1/2 and phi_3'(-1/2) = 1/2, so c = -1; asymptote x + 2y + 2 = 0. (Key point: Simple root)

3. At m = -1, the equation is -c^2 + 0 + 8 = 0. What is c?
   A. 8
   B. +-2
   C. +-4
   D. +-2 sqrt(2)
   Answer: D. c^2 = 8 gives c = +-2 sqrt 2. (Key point: Double root)

4. Which set lists all asymptotes?
   A. x + 2y + 2 = 0 and x + y = +-2 sqrt(2)
   B. x + 2y + 2 = 0 only
   C. x + y = 0 and x + 2y = 0
   D. x + y = +-8
   Answer: A. One simple-root line plus two parallel lines from the double root. (Key point: Final answer)

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