How to Identify the Four Conic Sections

When you slice up a cone, each conic section has its own standard form of an equation with x and y variables that you can graph on the coordinate plane. You can write the equation of a conic section if you are given key points on the graph, or you can graph the conic section from the equation. There are various ways that you can alter the shape of each of these graphs, but the general graph shapes still remain true to the type of curve that they are.

It is important to be able to identify which conic section is which by just the equation because sometimes that’s all you will be given (you won’t always be told what type of curve you are graphing). Certain key points are common to all conics (vertices, foci, and axes to name a few), so you start by plotting these key points and then identifying what kind of curve they form.

The above figure illustrates how a plane intersects the cones to create the conic sections.

The equations of conic sections are very important because they not only tell you which conic section you should be graphing, but they tell you what the graph should look like. There are trends in the appearance of each conic section based on the values of the constants in the equation. Usually these constants are referred to as a, b, h, v, f, and d. Not every conic will have all of these constants, but conics that do have them will be affected in the same way by changes in the same constant. Conic sections can come in all different shapes and sizes: big, small, fat, skinny, vertical, horizontal, and more.

An equation has to have x-squared and/or y-squared to create a conic. If neither x nor y is squared, then the equation will be of a line (not considered a conic section). None of the variables of a conic section may be raised to any power higher than two.

There are certain characteristics you will find unique to each type of conic that hint to you which of the conic sections you are graphing. In order to recognize these characteristics the way they are written, it is important that the x-squared term and the y-squared term are on the same side of the equal sign.

Circle

A circle is the set of all points a given distance (the radius, r) from a given point (the center). To get a circle from the right cones, the plane slice occurs parallel to the base of either cone, but does not slice through the element of the cones.

You can identify the equation for a circle when x and y are both squared, and the coefficients on them are the same — including the sign. For example, take a look at

Notice that the x-squared and y-squared have the same coefficient (positive 3). That’s all the info you need to recognize that you’re working with a circle.

Parabola

A parabola is a curve where every point on the curve is equidistant from one point (the focus) and a line (the directrix). It looks a lot like the letter U, although it may be upside down or sideways. To form a parabola, the plane slices through parallel to the side of the cones (any side works, but the bottom and top are forbidden).

You can identify the equation for a parabola when either x or y is squared — not both. For example, the equations

are both parabolas. In the first equation, you see an x-squared but no y-squared, and in the second equation, you see a y-squared but no x-squared. Nothing else matters — sign and coefficients will change the physical appearance of the parabola (which way it opens or how fat it is) but won’t change the fact that it’s a parabola.

Ellipse

An ellipse is the set of all points where the sum of the distances from two points (the foci) is constant, and you may be more familiar with the term oval. In order to get an ellipse from the two right cones, the plane must cut through one cone, not parallel to the base, and not through the element.

You can identify the equation for an ellipse when x and y are both squared and the coefficients are positive but different. For example, the equation

is one example of an ellipse. The coefficients on x-squared and y-squared are different, but both are positive.

Hyperbola

A hyperbola is the set of points where the difference of the distances between two points is constant. The shape of the hyperbola is difficult to describe without a picture, but it looks visually like two parabolas (although they are very different mathematically) mirroring one another with some space between the vertices. To get a hyperbola, the slice cuts the cones perpendicular to their bases (straight up and down), but not through the element.

You can identify the equation for a hyperbola when x and y are both squared and exactly one of the coefficients is negative (coefficients may be the same or different). For example, the equation

is an example of a hyperbola. This time, the coefficients on x-squared and y-squared are different, but one of them is negative, which is a requirement to get the graph of a hyperbola.