Conic Sections: Ellipses 

In this lesson you will learn how to write equations of ellipses and graphs of ellipses will be compared with their equations. Definition: An ellipse is all points found by keeping the sum of the distances from two points (each of which is called a focus of the ellipse) constant. The midpoint of the segment connecting the foci is the center of the ellipse. An ellipse can be formed by slicing a right circular cone with a plane traveling at an angle to the base of the cone. This effect can be seen in the following video and screen captures. Part I  Ellipses centered at the originExample #1: In our first example the constant distance mentioned above will be 10, one focus will be place at the point (0, 3) and one focus at the point (0, 3).The graph of our ellipse with these foci and center at the origin is shown below. Notice that the right graph shown above does a better job of showing the entire ellipse. It was generated by using EXCEL's chart feature. Note that it is sometimes not clear on a graphing calculator that the top half and bottom half of the ellipse are connected because of the way the calculator draws the top and bottom halves separately.
An equation of this ellipse can be found by using the distance formula to calculate the distance between a general point on the ellipse (x, y) to the two foci, (0, 3) and (0, 3). Since this total distance is 10, we have the equation Note that 10 is also the total distance from the top of the ellipse, through its center to the bottom. This is shown as the dark red line and is called the major axis.
After eliminating radicals in the above equation, and simplifying, we have
Note: a complete derivation of this equation is provided at the bottom of this lesson.
If we let a = 5 and b = 4, this equation can be written: . The important features are:  a = 5, the distance from the center to the vertices of the ellipse in the longest direction (up and down from the center). This is called the semimajor axis.
 b = 4, the distance from the center to the vertices of the ellipse in the shortest direction (left and right of the center). This is called the semiminor axis.
 is the distance from the center to each focus. Each focus is found on the major axis.
Example #2: Consider the graph below that shows the ellipse . This graph is from a graphing calculator screen. In this case, the major axis is horizontal, but a is still 5, b is still 4 and c is still 3. The ellipse is the same, but it is elongated horizontally rather than vertically.
The graph of using EXCEL's chart feature is shown below.
Using this as a model, other equations describing ellipses with centers at the origin can be written.
Example #3: If the horizontal distance from the center to the vertices is b = 3 and the vertical distance from the center to the vertices is a = 4, then the equation is Each focus is a distance of from the center. Note that each focus is found vertically from the center since in this case the ellipse is longer in the vertical direction. Two examples of the graph for this relation are shown below. Example #4: If the horizontal distance from the center to the vertices is 6 and the vertical distance to the vertices is 2 and the center is at the origin, we have . Each focus is a distance of from the center. Two graphs of this ellipse are shown below. Part II  Ellipses translated away for the centerExample #5: Suppose the center is not at the origin (0, 0) but is at some other point such as (2, 1). A graph of this ellipse requires us to remember how graphs are moved horizontally and vertically by a change in the equation. Using Example #1 above, we have . This will move the graph in our initial example 2 units right and 1 unit down. Below you can compare the new translated graph with the original. Note that a = 5, b = 4 and the major axis is vertical.
A more complete graph using EXCEL is shown below. Using this as a model, other equations describing ellipses with centers at (2, 1) can be written.  If a = 3 and b = 2, and the major axis is horizontal, the equation would be .
 If a = 3 and b = 2, and the major axis is vertical, the equation would be .
Part III  Summary In summary, the equation of an ellipse is written by using the standard formulas where  2a is the length of the major axis,
 2b is the length of the minor axis, and
 is the distance from the center (h , k) to each focus.
Use the standard equation form to determine the equation of each ellipse, the location of each focus, and the distance from the center to each focus 

Part IV  Writing equations in standard form
Writing an equation for an ellipse in standard form and getting a graph sometimes involves some algebra. For example, the equation is an equation of an ellipse. To see this, we will use the technique of completing the square.
Our first step will be to start with the equation . Completing the square for both x and y we have This is standard form of an ellipse with center (1, 4), a = 3, b = 2, and c = . Note that the major axis is vertical with one focus is at and other at
Part V  Graphing ellipses in standard form with a graphing calculator
To graph an ellipse in standard form, you must fist solve the equation for y. Two examples follow.  Given the equation , then
Enter this into the calculator as The calculator will graph the top and bottom halves of the ellipse using Y1 and Y2.  Given the equation , then
Enter this into the calculator as
. The calculator will graph the top and bottom halves of the ellipse using Y1 and Y2. Remember that the calculator does not join the top and bottom halves of the graph very well in this window. This is because of the restriction on the number of pixels plotted on the graphing calculator screen. As shown in many of the examples studies above, using EXCEL's chart feature will show the entire ellipse more clearly and completely. Put each equation into standard form, locate each focus, and determine the distance from the center to each focus 


Derivation:
We calculate the distance from the point on the ellipse (x, y) to the two foci, (0, 3) and (0, 3). This total distance is 10 in this example: which is the same as Square both sides: Expand terms and subtract x^{2} from both sides: Subtract y^{2}, 9, 6y, and 100 from both sides: Divide both sides by – 4: Square both sides: Subtract 9y^{2}, 150y, and 225 from both sides: Note that 5 is exactly half of 10, the constant distance, and 3 is the distance from the center to each focus. Let a = 5 and c = 3. Let b = 4.
This derivation can now be used as a model to make our work much easier. A standard form of the ellipse equation is where  a is the distance from the center to the endpoints (vertices) of the ellipse in the elongated direction (along the major axis) and
 b is the distance from the center to the vertices of the ellipse in the shorter direction (along the minor axis) and
 c is the distance from the center to each focus and .

M Ransom J Anderson


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