Conic Sections

Circles, ellipses, parabolas and hyperbolas are related curves called conic sections. A plane cutting a right circular double cone can produce these shapes. Their equations also arise from precise distance conditions in a coordinate plane.

Each conic can be represented by a second-degree equation in two coordinates. Recognizing a family is useful, but a family label alone is not enough: the equation may be shifted, rotated, degenerate or have no real points.

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Conic sections formed by cutting a double cone

A plane cutting one nappe, or half, of a double cone can produce an ellipse. A circle is the special case made by a plane perpendicular to the cone’s axis. A plane parallel to a generator, one of the straight lines along the cone, produces a parabola when the cut is non-degenerate. A plane cutting both nappes without passing through the vertex produces a hyperbola.

A perspective view of cone sections

The sections have different shapes even though they are formed from the same cone. Only finite portions of the unbounded curves are shown.

Diagram explanation and source

A cross-section perpendicular to the cone axis is circular; an oblique closed section is an ellipse; a section parallel to a generator is parabolic. The source shows one visible hyperbola branch on a portion of the cone. The complete non-degenerate hyperbola uses both nappes of the double cone and has two branches, shown in the coordinate comparison below. This perspective illustration is not a measured coordinate graph.

Adapted from Conic Sections.svg, by Magister Mathematicae (original); Phancy Physicist (English derivative); Glrx (internationalization). CC BY-SA 3.0. English-only labels and contrast adjustments; geometry unchanged. The inline adaptation retains the same licence.

The cone construction explains the name. For calculation, the distance definitions are often more useful. A circle fixes distance from one centre. A parabola equates distance from a focus with distance from a directrix. An ellipse fixes the sum of distances to two foci, and a hyperbola fixes the absolute difference.

Recognize standard coordinate forms

With the relevant centre or vertex at the origin, the main axis-aligned forms are:

Standard forms, before translations or rotations.
Curve	Standard form	Essential conditions
Circle	$x^2+y^2=r^2$	$r>0$
Ellipse	$x^2/a^2+y^2/b^2=1$	$a,b>0$ ; unequal axes for a non-circular ellipse
Parabola	$x^2=4py$	$p\ne0$
Hyperbola	$x^2/a^2-y^2/b^2=1$	$a,b>0$

Replacing $x$ by $x-h$ and $y$ by $y-k$ translates these forms. A horizontal parabola squares $y-k$ instead. A vertical hyperbola has the positive squared term in $y-k$ .

Compare the four conic families

Each equation specifies a different locus; the circle is a special ellipse.

Diagram explanation and exact data

The four windows use equal units. The parabola and hyperbola are only finite displayed portions of unbounded curves; the hyperbola has two separate branches. Exact sample points are provided below.

Compare the four conic families: exact reference data
Curve	Equation	Sample points
Circle	$x^2+y^2=4$	$(2,0),(0,2)$
Ellipse	$x^2/9+y^2/4=1$	$(3,0),(0,2)$
Parabola	$x^2=4y$	$(0,0),(2,1),(-2,1)$
Hyperbola	$x^2/4-y^2=1$	$(2,0),(-2,0)$

The signs, exponents and right-hand constant matter together. “Two squared terms means an ellipse” is not a safe rule: a difference of squares can describe a hyperbola or a pair of lines.

Simplify before classifying

Consider $4x^2+y^2-8x+4y=8$ . Group and complete the squares:

4(x^2-2x)+(y^2+4y)=8,

4(x-1)^2+(y+2)^2=16.

Divide by 16:

\frac{(x-1)^2}{4}+\frac{(y+2)^2}{16}=1.

This is a vertical ellipse centred at $(1,-2)$ , with horizontal semi-axis 2 and vertical semi-axis 4. It is not a circle because the squared-coordinate coefficients differ after normalization.

Compare $x^2-y^2=0$ . Factoring gives $(x-y)(x+y)=0$ , which is the union of the two intersecting lines $y=x$ and $y=-x$ . It is a degenerate conic, not a non-degenerate hyperbola.

The general second-degree equation

A general conic equation is

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

where at least one of $A,B,C$ is non-zero. The mixed term $Bxy$ can indicate rotation of the principal axes.

The discriminant of the quadratic part is

\Delta=B^2-4AC.

For a real, non-degenerate conic, $\Delta<0$ indicates ellipse type, including circles; $\Delta=0$ indicates parabola type; and $\Delta>0$ indicates hyperbola type. The qualification is essential: the sign does not by itself establish that a real non-degenerate curve exists. The OpenStax rotation-of-axes chapter explains this classification.

Examples of the test

For $x^2/9+y^2/4=1$ , $B=0$ , $A=1/9$ and $C=1/4$ , so $\Delta=-1/9<0$ : ellipse type.

For $x^2=8y$ , $A=1$ , $B=C=0$ , so $\Delta=0$ : parabola type.

For $xy=1$ , $B=1$ and $A=C=0$ , so $\Delta=1>0$ : a rotated hyperbola. Rotating axes can simplify its equation, but changing the coordinate labels does not change the curve’s underlying shape.

Degenerate and empty cases

Some second-degree equations describe smaller sets or no real set:

$x^2+y^2=0$ contains only $(0,0)$ .
$x^2+y^2=-1$ has no real solutions.
$x^2-y^2=0$ describes two intersecting lines.
$x^2=1$ describes two parallel lines, $x=1$ and $x=-1$ .

These examples show why completing squares, factoring or checking real solutions must accompany a discriminant test. An equation with an ellipse-type discriminant can still describe a point or an empty real set.

Distance conditions distinguish the curves

A circle has one constant radius. For a non-circular ellipse, the two foci lie inside the curve and their distance sum is $2a$ . For a hyperbola, the focal distance difference has magnitude $2a$ , and the foci lie beyond the vertices along the transverse axis.

A parabola has one focus and a directrix rather than a pair of foci at finite positions. Its points do not stay a fixed distance from the focus; they stay equally far from the focus and the line.

These are definitions, not merely visual similarities. An oval sketch is not necessarily an ellipse, and an arbitrary U-shaped curve is not necessarily a parabola.

Choose a useful next calculation

Once the family is established, identify the centre or vertex, the axis orientation and the relevant parameters. Then calculate intercepts, foci or asymptotes as the problem requires. Verify a few exact points against the original equation and use equal unit scales for geometric interpretation.

For an ellipse, compare squared semi-axis lengths before locating the foci. For a hyperbola, identify the positive squared term before finding its vertices and asymptotes. These different decisions are why classification should come before calculation.

Practice questions

Classify $x^2/9+y^2/4=1$ and identify its semi-axis lengths.
What set does $x^2-y^2=0$ describe?
Does $B^2-4AC<0$ guarantee a real ellipse? Give a counterexample.

Check your answers

It is an ellipse with horizontal semi-axis 3 and vertical semi-axis 2. The denominators are the squared semi-axis lengths.
Factoring gives $(x-y)(x+y)=0$ , so it is the union of $y=x$ and $y=-x$ .
No. For $x^2+y^2=-1$ , the discriminant is $-4$ , but no real points satisfy the equation.