---
title: "Asymptotes of y³+2xy²+x²y−y+1=0"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:asymptotes-cubic-y3"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/asymptotes-cubic-y3
site: BCT Engineering
---

# Asymptotes of y³+2xy²+x²y−y+1=0

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Asoj; 2073 Shrawan; 2071 Chaitra (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-cubic-y3
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Define asymptote to a curve. Find the asymptotes of the curve y³ + 2xy² + x²y − y + 1 = 0.

## Short answer

An asymptote is a line whose distance from a point on the curve tends to zero as the point tends to infinity. The slopes from φ₃(m)=m(m+1)²=0 give m=0, m=−1 (double). The asymptotes are y=0, x+y−1=0, x+y+1=0.

## Answer

#### Definition
An asymptote of a curve is a straight line such that the distance of a point P of the curve from the line tends to zero as P moves to infinity along some branch of the curve. (It is a tangent to the curve at infinity.)

#### Method
For an algebraic curve of degree n, write F = φₙ(x,y) + φₙ₋₁ + φₙ₋₂ + ... and put φₖ(m) = φₖ(1, m).

- Slopes m are the roots of φₙ(m) = 0.
- For a simple root, the intercept c is given by φₙ₋₁(m) + c φₙ′(m) = 0.
- For a repeated root (twice), c is given by c² φₙ″(m)/2! + c φₙ₋₁′(m) + φₙ₋₂(m) = 0.
- Parallel to the x-axis: coefficient of the highest power of x equated to zero.

#### Solution
F = y³ + 2xy² + x²y − y + 1 = 0, degree 3.

```text
φ₃(m) = m³ + 2m² + m = m(m+1)²
φ₂(m) = 0
φ₁(m) = −m
φ₀    = 1
```
φ₃(m) = 0 gives m = 0 (simple) and m = −1 (twice).

**m = 0:** φ₃′(0) = 1, φ₂(0) = 0, so c = 0. Asymptote: **y = 0**. (Also seen from the coefficient of x², which is y.)

**m = −1 (double):** φ₃″(m)/2 = 3m + 2 = −1 at m = −1; φ₂′ = 0; φ₁(−1) = 1.

−c² + 0 + 1 = 0, so c = ±1.

Asymptotes: **y = −x + 1** and **y = −x − 1**.

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

## Check yourself

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

1. An asymptote of a curve is a line such that:
   A. it touches the curve at one point
   B. it cuts the curve at two points
   C. it is parallel to the x-axis
   D. the distance from a point of the curve to the line tends to zero as the point goes to infinity along a branch
   Answer: D. An asymptote is a tangent at infinity: the distance tends to zero as P moves to infinity. (Key point: Definition)

2. For y^3 + 2xy^2 + x^2 y - y + 1 = 0, phi_3(m) = m^3 + 2m^2 + m. Which slopes result?
   A. m = 0 (simple), m = -1 (twice)
   B. m = 1, m = -1, m = 0
   C. m = 0 (twice), m = -1
   D. m = 0 only
   Answer: A. m(m+1)^2 = 0 gives m = 0 simple and m = -1 double. (Key point: Slopes)

3. For the repeated root m = -1, which equation gives c?
   A. c^2 phi_3''(m)/2 + c phi_2'(m) + phi_1(m) = 0
   B. phi_2(m) = c
   C. c phi_3''(m) = 0
   D. phi_3(m) + c phi_3'(m) = 0
   Answer: A. For a double root, the intercept equation involves phi_n''/2, phi_{n-1}' and phi_{n-2}. (Key point: Double root formula)

4. With m = -1, -c^2 + 1 = 0. What asymptotes follow?
   A. y = -x only
   B. y = x + 1 and y = x - 1
   C. y = -x + 1 and y = -x - 1
   D. y = 0 only
   Answer: C. c = +1 or -1 so the lines are y = -x + 1 and y = -x - 1; together with y = 0 these are the three asymptotes. (Key point: Result)

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