# Conics

## Ellipses


### 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.
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.
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.
We summarize the steps for reference.
Sometimes our equation will first need to be put in standard form.

### Find the Equation of an Ellipse with Center at the Origin

If we are given the graph of an ellipse, we can find the equation of the ellipse.

### Graph an Ellipse with Center Not at the Origin

The ellipses we have looked at so far have all been centered at the origin. We will now look at ellipses whose center is 
The equation is  and when  the major axis is horizontal so the distance from the center to the vertex is a. When  the major axis is vertical so the distance from the center to the vertex is b.
If we look at the equations of  and  we see that they are both ellipses with  and  So they will have the same size and shape. They are different in that they do not have the same center.

Notice in the graph above that we could have graphed  by translations. We moved the original ellipse to the right 3 units and then up 1 unit.
In the next example we will use the translation method to graph the ellipse.
When an equation has both an  and a  with different coefficients, we verify that it is an ellipsis by putting it in standard form. We will then be able to graph the equation.

### Solve Application with Ellipses

The orbits of the planets around the sun follow elliptical paths.

### Key Concepts

1. 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.


If we draw a line through the foci intersects the ellipse in two points—each 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.
2. Standard Form of the Equation an Ellipse with Center  The standard form of the equation of an ellipse with center  is

The x-intercepts are  and 
The y-intercepts are  and 
3. How to an Ellipse with Center 


4. Standard Form of the Equation an Ellipse with Center  The standard form of the equation of an ellipse with center  is

When  the major axis is horizontal so the distance from the center to the vertex is a.
When  the major axis is vertical so the distance from the center to the vertex is b.

### Practice Makes Perfect

Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
In the following exercises, graph each equation by translation.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
In the following exercises, graph the equation.
Solve Application with Ellipses

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?