Conic Sections — Circle, Ellipse, Parabola & Hyperbola

Conic sections are the curves called circle, ellipse, parabola, and hyperbola. In geometry, they can be formed by slicing a double cone with a plane. In algebra, they matter because their equations tell you the shape, center or vertex, and other key features.

If you need the quick version, use these definitions:

• A circle keeps every point the same distance from one center.
• An ellipse keeps the sum of distances to two fixed points constant.
• A parabola keeps each point the same distance from one focus and one directrix.
• A hyperbola keeps the absolute difference of distances to two fixed points constant.

Why circle, ellipse, parabola, and hyperbola are one family

The word "conic" comes from cone. When a plane cuts a double cone at different angles, the intersection can produce these different curves. A circle is a special case of an ellipse, which is why some books group it under the ellipse family and some list it separately.

There is also a unifying focus-directrix view using eccentricity $e$:

• circle: special ellipse case with $e = 0$
• ellipse: $0 < e < 1$
• parabola: $e = 1$
• hyperbola: $e > 1$

You do not need eccentricity to solve basic problems, but it helps explain why the four shapes are part of one family instead of four unrelated topics.

How to identify a conic from its equation

In beginner coordinate-geometry problems, once the equation has been simplified into a standard form, these clues usually work:

• Circle: both squared terms appear with the same coefficient after scaling, such as $x^2 + y^2 = 25$.
• Ellipse: both squared terms appear with the same sign but different positive coefficients in standard form, such as $\frac{x^2}{9} + \frac{y^2}{4} = 1$.
• Parabola: only one variable is squared in the standard orientation forms, such as $y = x^2$ or $x = y^2$.
• Hyperbola: the squared terms have opposite signs, such as $\frac{x^2}{9} - \frac{y^2}{4} = 1$.

That shortcut only works after the equation is cleaned up. If the terms are expanded or the graph is shifted, collect like terms and complete the square first.

One worked example

Classify the curve

$$4x^2 + 9y^2 = 36$$

First divide both sides by $36$:

$$\frac{4x^2}{36} + \frac{9y^2}{36} = 1$$

which simplifies to

$$\frac{x^2}{9} + \frac{y^2}{4} = 1$$

Now the pattern is clear:

• both squared terms are present
• both have positive signs
• the denominators are different

So this is an ellipse, not a circle. Its center is at the origin, its horizontal semi-axis is $3$, and its vertical semi-axis is $2$.

This is the main move in many conic sections problems. Rewrite first, classify second.

What each conic means

Circle

A circle is the set of all points a fixed distance from a center. In the standard form

$$(x-h)^2 + (y-k)^2 = r^2$$

the center is $(h,k)$ and the radius is $r$, with the condition $r \ge 0$.

Ellipse

An ellipse is the set of all points whose distances to two fixed points, called foci, add to a constant. In standard position, a common form is

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

with $a > 0$ and $b > 0$. It looks like a stretched circle, but the two-focus definition is the important geometric idea.

Parabola

A parabola is the set of all points equally distant from a focus and a directrix. A common standard form is

$$(x-h)^2 = 4p(y-k)$$

and the sideways version is

$$(y-k)^2 = 4p(x-h)$$

The value $p$ controls how far the focus is from the vertex and which way the graph opens.

Hyperbola

A hyperbola is the set of all points for which the absolute difference of distances to two foci stays constant. In standard position, one form is

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Its two branches and asymptotes come from that distance condition.

Common conic section mistakes

Treating every quadratic graph as a parabola

A parabola is only one kind of conic. If both $x^2$ and $y^2$ appear, you should stop and check whether the graph is actually a circle, ellipse, or hyperbola.

Classifying too early

An expanded equation can hide the shape. For example, a circle may not look like a circle until you complete the square. Classification is much safer after rewriting.

Forgetting that a circle is a special ellipse

In many school problems, circle is listed separately because it is simple and common. That is useful, but geometrically it still belongs to the conic family.

Mixing up the focus definitions

The ellipse uses a sum of distances. The hyperbola uses an absolute difference. The parabola compares distance to a focus with distance to a directrix, not with distance to a second focus.

Where conic sections are used

Conics show up whenever geometry depends on distance rules or second-degree equations. Circles appear in basic geometry and symmetry. Ellipses appear in idealized orbital models. Parabolas appear in reflective geometry and in projectile models when air resistance is neglected. Hyperbolas appear in some navigation and signal-location models that depend on differences in distance or arrival time.

Even if you never use the cone picture again, conics matter because they train you to connect an equation with a shape and a shape with a geometric rule.

Try a similar problem

Take the equation

$$x^2 + y^2 - 6x + 2y - 6 = 0$$

and rewrite it by completing the square before you classify it. That is a good next step because it forces you to use the main habit that makes conic sections much easier: do not guess from the raw equation when a cleaner form is available.