Conics
Ellipses
Learning Objectives
By the end of this section, you will be able to:
- Graph an ellipse with center at the origin
- Find the equation of an ellipse with center at the origin
- Graph an ellipse with center not at the origin
- Solve application with ellipses
Before you get started, take this readiness quiz.
Graph an Ellipse with Center at the Origin
The next conic section we will look at is an ellipse. We define an ellipse as all points in a plane where the sum of the distances from two fixed points is constant. Each of the given points is called a focus of the ellipse.
An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse.
We can draw an ellipse by taking some fixed length of flexible string and attaching the ends to two thumbtacks. We use a pen to pull the string taut and rotate it around the two thumbtacks. The figure that results is an ellipse.
A line drawn through the foci intersect the ellipse in two points. Each point is called a vertex of the ellipse. The segment connecting the vertices is called the major axis. The midpoint of the segment is called the center of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.
We mentioned earlier that our goal is to connect the geometry of a conic with algebra. Placing the ellipse on a rectangular coordinate system gives us that opportunity. In the figure, we placed the ellipse so the foci are on the x-axis and the center is the origin.
The definition states the sum of the distance from the foci to a point is constant. So is a constant that we will call so, We will use the distance formula to lead us to an algebraic formula for an ellipse.
To graph the ellipse, it will be helpful to know the intercepts. We will find the x-intercepts and y-intercepts using the formula.
The standard form of the equation of an ellipse with center is
The x-intercepts are and
The y-intercepts are and
Notice that when the major axis is horizontal, the value of a will be greater than the value of b and when the major axis is vertical, the value of b will be greater than the value of a. We will use this information to graph an ellipse that is centered at the origin.
| Ellipse with Center | ||
|---|---|---|
| Major axis | on the x– axis. | on the y-axis. |
| x-intercepts | ||
| y-intercepts |
Graph:
Graph:
Graph:
We summarize the steps for reference.
- Write the equation in standard form.
- Determine whether the major axis is horizontal or vertical.
- Find the endpoints of the major axis.
- Find the endpoints of the minor axis
- Sketch the ellipse.
Sometimes our equation will first need to be put in standard form.
Graph
| We recognize this as the equation of an ellipse since both the x and y terms are squared and have different coefficients. |
|
| To get the equation in standard form, divide both sides by 16