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Ellipse Equation

The standard form of an ellipse equation is as follows:

An ellipse is an oval-shaped curve that looks like a stretched-out circle. Ellipses have two foci, and every point on the ellipse has the same total distance to both foci.

If $a>b$, the ellipse is wider and stretches horizontally. If $b>a$, the ellipse is taller and stretches vertically. The longer direction is called the major axis, and the shorter one is the minor axis.

If the larger number is under $(x-h)^2$, the ellipse is wider. If the larger number is under $(y-k)^2$, it is taller.

To graph an ellipse, start at the center and move out $a$ and $b$ units in the right directions.

Ellipse

In a previous lesson, we learned about conics. Conics are shapes that are formed when a flat surface, called a plane, cuts through a circular cone.

One type of conic is an ellipse.

Ellipses have two axes of symmetry. The major axis is the longer one. The minor axis is the shorter one.

Most of the ellipses you will see in math are centered at the origin $(0,0)$. For these ellipses, there is a standard form of the equation that helps us describe their shape clearly and neatly.

The Standard Form

You won't need to know how to derive this formula, but recognize that mathematicians used the distance formula to get here.

The Standard/General Form of an Ellipse with the major axis on the x-axis is:

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

, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. $(\pm a, o)$ and $(0, \pm b)$ are coordinates of the vertices.

The Standard/General Form of an Ellipse with the major axis on the y-axis is:

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

, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. $(\pm a, o)$ and $(0, \pm b)$ are coordinates of the vertices.

As we can see from the above image, the length of the major axis is $2a$, and the length of the minor axis is $2b$. Imagine that the center of the ellipse is at the origin, $(0,0)$.

The major points on the ellipse are where the axes intersect with the ellipse: $(-a,0)$, $(0,b)$, $(a,0)$, and $(0, -b)$.

(Video) Standard Form of an Ellipse

by Brian McLogan

In this video by Brian McLogan, you will learn about an ellipse's standard form, which will help you to graph it.

Summary

The video begins with explaining why an ellipse can be hard to graph when its equation is not in standard form. You will then see an example on how to group the $x$-terms and $y$-terms to make the equation easier to work with.

Next, you will learn how to factor and complete the square for both variables. The video explains how to keep the equation balanced when you add numbers while completing the square. You will then see how to rewrite the new expressions as squared binomials.

Finally, the video will show you how to make the equation equal to $1$ so that the ellipse is written in standard form.