# Analytic Geometry

## The Ellipse


### Learning Objectives

1. Complete the square of a binomial expression. (IA 9.2.1)
2. Graph a circle. (IA 11.1.4)

### Objective 1: Complete the square of a binomial expression. (IA 9.2.1)

But what happens if we have to solve an equation where the trinomial is not a perfect square? 
For example,  ? For these types of equations, we can use a process called completing the square.
Recall  .
We can use the Binomial Squares Pattern to make a perfect square. 

### Practice Makes Perfect 

Determine what number would have to be added to the given terms to create a perfect square trinomial. Then rewrite as a binomial squared.

### Objective 2: Graph a circle. (IA 11.1.4)


A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, (h, k) and the fixed distance is called the radius, r, of the circle.

The standard or graphing form of the equation of a circle with center, (h, k) and radius, r, is  .
The general form of the equation of a circle is  . If we are given an equation in general form, we can change it to standard, also called the graphing form, by completing the squares in both x  and y. Then we can graph the circle using its center and radius. 

### Practice Makes Perfect

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in , is such a room.Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014. It is an semi-circular room called a whispering chamber because the shape makes it possible for sound to travel along the walls and dome. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

### Writing Equations of Ellipses in Standard Form

A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in .
Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points 
  in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See .
Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See .
In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.
To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

### Deriving the Equation of an Ellipse Centered at the Origin

To derive the equation of an ellipse centered at the origin, we begin with the foci 
  and 
  The ellipse is the set of all points 
  such that the sum of the distances from 
  to the foci is constant, as shown in .
If 
  is a vertex of the ellipse, the distance from 
  to 
  is 
  The distance from 
  to 
  is 
  . The sum of the distances from the foci to the vertex is
If 
  is a point on the ellipse, then we can define the following variables:
By the definition of an ellipse, 
  is constant for any point 
  on the ellipse. We know that the sum of these distances is 
  for the vertex 
  It follows that 
  for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.
Thus, the standard equation of an ellipse is 
  This equation defines an ellipse centered at the origin. If 
  the ellipse is stretched further in the horizontal direction, and if 
  the ellipse is stretched further in the vertical direction.

### Writing Equations of Ellipses Centered at the Origin in Standard Form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.
The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

### Writing Equations of Ellipses Not Centered at the Origin

Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated 
  units horizontally and 
  units vertically, the center of the ellipse will be 
  This translation results in the standard form of the equation we saw previously, with 
  replaced by 
  and y replaced by 
 


###  Graphing Ellipses Centered at the Origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form 
  for horizontal ellipses and 
  for vertical ellipses.

### Graphing Ellipses Not Centered at the Origin

When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point, 
    we use the standard forms 
    for horizontal ellipses and 
    for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

### Solving Applied Problems Involving Ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See . In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.  When these chambers are placed in unexpected places, such as the ones inside Bush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions among travelers. 

### Key Equations


### Key Concepts

1. An ellipse is the set of all points 
   in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
2. When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See  and .
3. When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See  and .
4. When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See  and .
5. Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.
For the following exercises, find the foci for the given ellipses.

### Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.
For the following exercises, use the given information about the graph of each ellipse to determine its equation.
For the following exercises, given the graph of the ellipse, determine its equation.

###  Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula 
     
    

###  Real-World Applications
