Asymptotes

At a Glance

• Frequency: 1 sub-part across 1 of 13 years (2020)
• Priority tier: T4
• Marks (count): 10 (1)
• Average solve time: ~10 min
• Difficulty mix: medium 1
• Section: A | Dominant type: computation

Why This Chapter Matters

Asymptotes appeared once in 2020 as a 10-mark Section A computation, making it a low-frequency atom. UPSC typically presents an algebraic curve (cubic or rational) and asks you to find all asymptotes — vertical, horizontal, and oblique. The topic is self-contained and procedural: the three-type framework below covers every case. It also reinforces limit-at-infinity reasoning used across many other calculus atoms.

Minimum Theory

Types of Asymptotes

Vertical asymptote at $x = a$: occurs when the denominator of a rational expression vanishes at $x = a$ (and the numerator does not), i.e., $\lim_{x \to a} |f(x)| = \infty$.

Horizontal asymptote $y = L$: occurs when $\lim_{x \to \pm\infty} f(x) = L$ (finite).

Oblique (slant) asymptote $y = mx + c$: occurs when

$m = \lim_{x \to \pm\infty} \frac{f(x)}{x}, \qquad c = \lim_{x \to \pm\infty} \bigl(f(x) - mx\bigr).$

If $m = 0$ and the limit for $c$ is finite, the result is a horizontal asymptote.

Asymptotes of Implicit Algebraic Curves

For a curve $F(x, y) = 0$ of degree $n$, the oblique asymptotes $y = mx + c$ are found by:

1. Finding slopes $m$: Substitute $y = mx + c$ into $F$ and collect by degree. The terms of degree $n$ give $\phi_n(m) = 0$, where $\phi_n(m)$ is the leading homogeneous part evaluated at $(1, m)$. Each root $m$ is a potential asymptote slope.

2. Finding intercepts $c$: For each root $m$ of $\phi_n(m) = 0$, the intercept satisfies

$c = -\frac{\phi_{n-1}(m)}{\phi_n'(m)},$

where $\phi_{n-1}(m)$ is the homogeneous part of degree $n-1$ evaluated at $(1, m)$, and $\phi_n'(m) = d\phi_n/dm$.

3. Vertical asymptotes: arise when $\phi_n(m)$ has infinite roots, i.e., the coefficient of $y^n$ in the highest-degree terms is zero — investigate directly.

Asymptote Count Rule

A curve of degree $n$ has at most $n$ asymptotes (real). Some may coincide or be complex (discarded).

Question Archetypes

all-asymptotes-algebraic (1 question; 2020)

Recognition Cues

• The curve is given as an implicit polynomial equation $F(x,y) = 0$ of degree 3 or 4
• The question says “find all asymptotes” — all three types must be checked
• 10 marks suggests the curve has 2–3 distinct asymptotes requiring full working
• Oblique asymptotes require the $\phi_n$/$\phi_{n-1}$ method, not just limit reasoning

Solution Template

1. Write the degree-$n$ homogeneous part $\phi_n(m) = F_n(1, m)$ and solve $\phi_n(m) = 0$ for $m$.
2. For each finite $m$: compute $c = -\phi_{n-1}(m)/\phi_n'(m)$; asymptote is $y = mx + c$.
3. Check for vertical asymptotes: if $\phi_n(m)$ has a root at $m = \infty$ (i.e., leading $y^n$ coefficient in $F$ is zero), find $x =$ const directly.
4. State all asymptotes clearly.

Worked Example

2020 Paper 1, 2020-P1-Q3b (10 marks)

Find all the asymptotes of the curve $y^3 - xy^2 - x^2y + x^3 + x^2 - y^2 = 0$.

Step 1. Identify degree and homogeneous parts.

The curve is degree 3. Group by degree:

• Degree-3 part: $y^3 - xy^2 - x^2y + x^3$.
• Degree-2 part: $x^2 - y^2$.

Step 2. Compute $\phi_3(m)$.

Set $x = 1$, $y = m$ in the degree-3 part:

$\phi_3(m) = m^3 - m^2 - m + 1 = (m-1)^2(m+1).$

Roots: $m = 1$ (double), $m = -1$.

Step 3. Compute intercepts.

$\phi_3'(m) = 3m^2 - 2m - 1.$

Degree-2 part at $(1, m)$: $\phi_2(m) = 1 - m^2$.

For $m = 1$: $\phi_3'(1) = 3 - 2 - 1 = 0$. Since both $\phi_3(1) = 0$ and $\phi_3'(1) = 0$, the asymptote $y = x + c$ requires a second-order treatment.

For $m = -1$: $\phi_3'(-1) = 3 + 2 - 1 = 4$.

$c = -\frac{\phi_2(-1)}{\phi_3'(-1)} = -\frac{1 - 1}{4} = 0.$

Asymptote: $y = -x$.

Step 4. Handle the double root $m = 1$.

When $\phi_3'(m) = 0$ at a root, expand $F(x, mx+c)$ and collect the coefficient of $x^{n-1}$ (here $x^2$) to find $c$, then the coefficient of $x^{n-2}$ to find a second asymptote or confirm the root yields only one asymptote.

Substitute $y = x + c$ into the full curve:

$F(x, x+c) = (x+c)^3 - x(x+c)^2 - x^2(x+c) + x^3 + x^2 - (x+c)^2.$

Expand:

• $(x+c)^3 = x^3 + 3cx^2 + 3c^2x + c^3$
• $x(x+c)^2 = x^3 + 2cx^2 + c^2x$
• $x^2(x+c) = x^3 + cx^2$
• $(x+c)^2 = x^2 + 2cx + c^2$

Collecting:

$F = (x^3 + 3cx^2 + \cdots) - (x^3 + 2cx^2 + \cdots) - (x^3 + cx^2 + \cdots) + x^3 + x^2 - x^2 - 2cx - c^2.$

Coefficient of $x^3$: $1 - 1 - 1 + 1 = 0$. Good (confirms $m=1$ is a root).

Coefficient of $x^2$: $3c - 2c - c + 1 - 1 = 0$. This is $0 = 0$ — indeterminate, confirming the double root means no unique $c$ from the $x^2$ equation alone.

Coefficient of $x^1$: $3c^2 - c^2 - 2c - 0 = 2c^2 - 2c$.

Set equal to zero: $2c(c - 1) = 0$, giving $c = 0$ or $c = 1$.

Two asymptotes from the double root: $y = x$ and $y = x + 1$.

Step 5. Check for vertical asymptotes.

The leading term in $y^3$ is present (coefficient 1 $\neq$ 0), so no vertical asymptote at $x = \infty$.

Step 6. State all asymptotes.

$\boxed{y = x, \quad y = x + 1, \quad y = -x}$

Common Traps

• Forgetting the double-root case: When $\phi_n'(m) = 0$ at a root of $\phi_n$, the standard formula $c = -\phi_{n-1}/\phi_n'$ blows up — you must substitute and expand to find $c$ values.
• Sign error in the $c$ formula: The formula is $c = -\phi_{n-1}(m)/\phi_n'(m)$ — the negative sign is easy to drop.
• Missing vertical asymptotes: Always check whether the $y^n$ coefficient vanishes; if it does, there may be a vertical asymptote not captured by the oblique method.
• Complex roots: Roots of $\phi_n(m) = 0$ may be complex; discard these (UPSC only expects real asymptotes).

Marks-Aware Writing

For a 10-mark computation:

• 2 marks — correctly identify and write the homogeneous parts $\phi_n$ and $\phi_{n-1}$.
• 3 marks — solve $\phi_n(m) = 0$, recognise any repeated roots, handle them correctly.
• 3 marks — compute $c$ for each root (including the expansion method for repeated roots).
• 2 marks — state all asymptotes clearly, one per line.

Label each asymptote as oblique/horizontal/vertical in your answer — it signals to the examiner that you understand the framework.

Practice Set

Only one historical question on this atom (shown above).