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

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

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2073 Chaitra; 2068 Shrawan (asked 2 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-x2-minus-y2-times-linear
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

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

## Short answer

The asymptotes are y = x, y = -x, and x + 2y + 1 = 0. They are obtained by setting φ₃(m)=0 giving slopes m=1,-1,-1/2 and computing c from c = -φ₂(m)/φ₃'(m).

## Answer

Curve: (x² − y²)(x + 2y + 1) + x + y + 1 = 0. Degree 3.

```text
φ₃(m) = (1 − m²)(1 + 2m)
φ₂(m) = (1 − m²)
φ₁(m) = 1 + m
```
φ₃(m) = 0 gives three simple slopes m = 1, m = −1, m = −1/2. For simple roots c = −φ₂(m)/φ₃′(m), where

φ₃′(m) = −2m(1 + 2m) + 2(1 − m²).

- **m = 1:** φ₂(1) = 0, so c = 0. Asymptote: **y = x**.
- **m = −1:** φ₂(−1) = 0, so c = 0. Asymptote: **y = −x**.
- **m = −1/2:** φ₂ = 3/4, φ₃′ = 0 + 2(3/4) = 3/2, so c = −(3/4)/(3/2) = −1/2. Asymptote: y = −x/2 − 1/2, i.e. **x + 2y + 1 = 0**.

**Answer: y = x, y = −x, x + 2y + 1 = 0.**

## Check yourself

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

1. phi_3(m) = (1-m^2)(1+2m) = 0 gives:
   A. m = 1, -1, 1/2
   B. m = 1, -1, -1/2 (all simple)
   C. m = 0, 1
   D. m = 1 twice and -1/2
   Answer: B. Three distinct simple roots. (Key point: Slopes)

2. For a simple root, c is given by:
   A. c = phi_3(m)/phi_2(m)
   B. c = phi_2(m) * phi_3'(m)
   C. c = -phi_1/phi_3
   D. c = -phi_2(m)/phi_3'(m)
   Answer: D. From phi_3'(m) c + phi_2(m) = 0. (Key point: Intercept formula)

3. Since phi_2(+-1) = 0, the asymptotes for m = +-1 are:
   A. y = 0
   B. y = x + 1 and y = -x + 1
   C. y = x and y = -x
   D. y = x - 1 only
   Answer: C. c = 0 for both. (Key point: m = 1 and -1)

4. The third asymptote is:
   A. y = -x/2
   B. 2x + y + 1 = 0
   C. x + 2y - 1 = 0
   D. x + 2y + 1 = 0
   Answer: D. c = -1/2, y = -x/2 - 1/2. (Key point: m = -1/2)

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