Math Core

Circles, ellipses, parabolas and hyperbolas look very different, but they belong to one family. They are all the curves you get by slicing a cone with a plane, and they all come from second-degree equations in $x$ and $y$ . Given an expanded equation, you can tell which conic it is just by looking at a few coefficients, and then complete the square to confirm the details.

Slicing a double cone

Picture two cones joined at their tips, like an hourglass that goes on forever. Cut them with a flat plane:

A plane perpendicular to the axis cuts a circle.
Tilt the plane a little and the circle stretches into an ellipse.
Tilt it until it is parallel to one edge of the cone and the curve opens up into a parabola.
Tilt it further so it cuts both halves of the cone and you get the two branches of a hyperbola.

If the plane passes through the tip, you get a degenerate conic: a single point, a single line, or a pair of crossing lines.

The general second-degree equation

Every conic can be written as

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,

where $A$ , $B$ and $C$ are not all zero. For the rest of this section, assume there is no $xy$ -term ( $B = 0$ ). Then the axes of the conic are horizontal and vertical, exactly like the standard forms you've studied, and the squared terms decide the type.

Classifying Ax² + Cy² + Dx + Ey + F = 0

Condition on $A$ and $C$	Conic	Example
$A = C$	circle	$2x^2 + 2y^2 - 8 = 0$
$A$ and $C$ have the same sign, $A \ne C$	ellipse	$x^2 + 4y^2 - 4 = 0$
$A$ and $C$ have opposite signs	hyperbola	$x^2 - 4y^2 - 4 = 0$
exactly one of $A$ , $C$ is $0$	parabola	$x^2 - 4y = 0$

A quick way to remember: look at the product $AC$ . Positive means circle or ellipse, negative means hyperbola, and zero means parabola.

The linear terms $Dx$ and $Ey$ only move the center or vertex. They never change the type of curve.

Worked example: Classify by inspection

Identify each conic.

(a) $3x^2 + 5y^2 - 12x + 10y - 8 = 0$

(b) $4x^2 - y^2 + 8x + 6y - 9 = 0$

(d) $7x^2 + 7y^2 - 14x + 21y = 0$

Solution.

(a) $A = 3$ and $C = 5$ have the same sign but are different: ellipse.

(b) $A = 4$ and $C = -1$ have opposite signs: hyperbola.

(d) $A = C = 7$ : circle.

Confirming with standard form

Classifying tells you the type. Completing the square tells you everything else: center, radius, vertices, foci. For a circle, the standard form is $(x - h)^2 + (y - k)^2 = r^2$ .

Worked example: A circle in disguise

Identify $x^2 + y^2 - 6x + 4y - 12 = 0$ and find its key features.

Solution. $A = C = 1$ , so it's a circle. Complete the square in both variables:

\begin{aligned} (x^2 - 6x + 9) + (y^2 + 4y + 4) &= 12 + 9 + 4 \\ (x - 3)^2 + (y + 2)^2 &= 25. \end{aligned}

The center is $(3, -2)$ and the radius is $5$ .

(x − 3)² + (y + 2)² = 25: a circle of radius 5.Open in grapher →

Degenerate cases

The coefficient test assumes the equation actually has a graph that is a curve. Sometimes completing the square reveals something else.

Worked example: When the ellipse collapses

Identify the graph of $x^2 + 4y^2 - 2x + 8y + 5 = 0$ .

Solution. $A = 1$ and $C = 4$ have the same sign, so the test says ellipse. Complete the square to check:

\begin{aligned} (x^2 - 2x + 1) + 4(y^2 + 2y + 1) &= -5 + 1 + 4 \\ (x - 1)^2 + 4(y + 1)^2 &= 0. \end{aligned}

A sum of two squares is $0$ only when both are $0$ . The only solution is $x = 1$ , $y = -1$ . The graph is the single point $(1, -1)$ , a degenerate ellipse.

Common mistake

Always finish by completing the square before you describe a graph. After completing the square, check the constant on the right side:

For an ellipse or circle, a right side of $0$ gives a single point, and a negative right side gives no graph at all. For instance $x^2 + y^2 = -4$ has no real solutions.
For a hyperbola, a right side of $0$ gives two crossing lines. For instance $x^2 - y^2 = 0$ is the pair of lines $y = x$ and $y = -x$ .

Conics with an xy-term

When $B \ne 0$ , the conic is rotated: its axes are tilted instead of horizontal and vertical. The sign test on $AC$ no longer works by itself, but a similar test does.

The discriminant test

For $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ (non-degenerate), compute $B^2 - 4AC$ :

$B^2 - 4AC < 0$ : ellipse (a circle when $B = 0$ and $A = C$ )
$B^2 - 4AC = 0$ : parabola
$B^2 - 4AC > 0$ : hyperbola

When $B = 0$ , the discriminant is just $-4AC$ , so this is the same as the $AC$ test from before.

Worked example: Rotated conics

Identify (a) $x^2 + 4xy + y^2 = 6$ and (b) $x^2 - xy + y^2 = 3$ .

Solution.

(a) $A = 1$ , $B = 4$ , $C = 1$ : $B^2 - 4AC = 16 - 4 = 12 > 0$ . It is a hyperbola, even though both squared terms are positive. The $xy$ -term changes everything.

(b) $A = 1$ , $B = -1$ , $C = 1$ : $B^2 - 4AC = 1 - 4 = -3 < 0$ . It is an ellipse, tilted at $45^\circ$ .

The hyperbola x² + 4xy + y² = 6 and the tilted ellipse x² − xy + y² = 3.Open in grapher →

Eccentricity ties the family together

Every non-degenerate conic has an eccentricity $e$ , and its value names the type: a circle has $e = 0$ , an ellipse has $0 < e < 1$ , a parabola has $e = 1$ , and a hyperbola has $e > 1$ . For an ellipse or hyperbola, $e = \dfrac{c}{a}$ . As a hyperbola's eccentricity grows, its branches open wider.

Tip

To classify in one glance, cover up everything except the $x^2$ , $xy$ and $y^2$ terms. The linear terms and the constant never change the type (though the constant can make the graph degenerate).

Practice

Practice 1

Identify the conic: $4x^2 + 4y^2 - 8x + 16y - 5 = 0$ . (Type circle, ellipse, parabola or hyperbola.)

Type your answer

Practice 2

Identify the conic: $2x^2 - 3y^2 + 4x + 6y - 7 = 0$ . (Type circle, ellipse, parabola or hyperbola.)

Type your answer

Practice 3

What is the graph of $5x^2 + 2y^2 - 20x + 4y + 12 = 0$ ?

Practice 4

Identify the conic: $3y^2 + 12y - x + 10 = 0$ . (Type circle, ellipse, parabola or hyperbola.)

Type your answer

Practice 5

Find the radius of the circle $x^2 + y^2 + 10x - 2y + 1 = 0$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Use the discriminant to identify $3x^2 - 2xy + 5y^2 - 7 = 0$ .

Practice 7

What is the graph of $9x^2 - y^2 = 0$ ?

Practice 8

For what value of $k$ is the graph of $kx^2 + 4xy + y^2 + 3x - 5 = 0$ a parabola?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.