Identifying Conic Sections from Their Equations

Name any conic at a glance. Count the squared terms, check the signs, compare the denominators — and read off the direction it opens. Covers the circle, parabola, ellipse and hyperbola side by side.

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Circle, parabola, ellipse, hyperbola — four curves, four standard equations. You do not need to plot anything to tell them apart. Look at how many squared terms there are, what signs they carry, and what sits on the right-hand side. Those three checks identify the curve every time, and a fourth tells you which way it opens.

Concept The circle

x^2 + y^2 = r^2

r
Both terms are squared and both are positive.
The coefficients of x^2 and y^2 are equal.
The right-hand side is r^2 , not r : if r^2 = 25 then r = 5 , and if r^2 = 9 then r = 3 .

A circle has no horizontal or vertical version — it looks the same in every direction, so rotating it changes nothing. That equal-coefficient test is what separates it from an ellipse.

Concept The parabola

x^2 = 4cy \qquad \text{or} \qquad y^2 = 4cx

The parabola is the odd one out: only one variable is squared. The other appears to the first power, carrying the coefficient 4c . If you see exactly one squared term, the curve is a parabola — no further test needed.

x² = 4cy x² = −4cy y² = 4cx y² = −4cx
x^2 squared → opens up or down.
y^2 squared → opens right or left.
A minus sign flips it to the opposite direction.
Concept The ellipse

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Both terms are squared and both are positive — as in a circle.
The right-hand side is 1.
The denominators are different. The larger is a^2 , the smaller b^2 .

The larger denominator points along the major axis: a^2 under x^2 gives a horizontal major axis, and a^2 under y^2 gives a vertical one.

There is a special case worth knowing. If the denominators become equal, so a^2 = b^2 , there is no longer a larger and a smaller axis — and the ellipse becomes a circle.

Concept The hyperbola

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \qquad \text{or} \qquad \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

The right-hand side is 1.
There is one positive term and one negative term — this is the giveaway.
a^2 always sits under the positive term.
b^2 always sits under the negative term.

Despite the curve being called a hyperbola — a name suggesting excess — its equation is the one containing a minus sign. That contrast is the easiest way to remember it.

Positive term is x^2 → opens horizontally, left and right, like a C and a reversed C.
Positive term is y^2 → opens vertically, up and down, like a U and an inverted U.

The bigger the gap between a and b , the wider the branches spread. And unlike the ellipse, setting a = b does not turn a hyperbola into a circle — it stays a hyperbola.

Note The quick comparison
Circle — x^2 + y^2 = r^2 ; both positive, equal coefficients; no direction.
Parabola — only one squared term; direction from which variable is squared and the sign.
Ellipse — both positive, different denominators, right side 1; larger denominator marks the major axis.
Hyperbola — one positive and one negative term, right side 1; the positive term decides the opening.

Work down the list in order and the curve names itself. Count the squared terms first, then check the signs, then compare the denominators.

Example Naming a curve and its direction

Identify the curve given by \dfrac{y^2}{25} - \dfrac{x^2}{9} = 1 , and say which way it opens.

Both variables are squared, so it is not a parabola.
One term is positive and the other negative, and the right side is 1 — that is a hyperbola.
The positive term is y^2 , so it opens vertically.
a^2 sits under the positive term: a^2 = 25 , and b^2 = 9 .
⟹ a vertical hyperbola, opening up and down, with a = 5 and b = 3
Example Telling an ellipse from a circle

Identify \dfrac{x^2}{16} + \dfrac{y^2}{4} = 1 and then \dfrac{x^2}{9} + \dfrac{y^2}{9} = 1 .

The first has both terms positive, right side 1, and denominators 16 and 4 — different, so it is an ellipse.
The larger denominator, 16, sits under x^2 , so the major axis is horizontal.
The second has both denominators equal to 9, so there is no major or minor axis.
Multiplying through by 9 turns it into x^2 + y^2 = 9 .
⟹ the first is a horizontal ellipse; the second is a circle of radius 3
Note Mistakes to avoid
Reading the right-hand side of a circle as r instead of r^2 .
Calling an ellipse a circle without checking that the denominators are actually equal.
Assuming a^2 is the denominator under x^2 — for a hyperbola it is whichever sits under the positive term.
Expecting a hyperbola with a = b to become a circle; only the ellipse does that.
Missing that a single squared term already settles the answer as a parabola.
Forgetting that a minus sign in front reverses a parabola's opening direction.
Summary
  1. One squared term means a parabola; the squared variable and the sign give the opening direction.
  2. Two positive squared terms with equal coefficients give a circle, whose right-hand side is r² rather than r.
  3. Two positive squared terms over different denominators, equal to 1, give an ellipse; the larger denominator lies along the major axis.
  4. One positive and one negative term equal to 1 gives a hyperbola, with a² under the positive term and b² under the negative one.
  5. Equal denominators turn an ellipse into a circle, but a hyperbola with a = b remains a hyperbola.
231. Arithmetic Sequences and Series
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