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High School Geometry: Help and Review13 chapters | 162 lessons

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Lesson Transcript

Instructor:
*Joshua White*

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson will cover the definition of ellipses and the standard form equation of an ellipse. It will also examine how to determine the orientation of an ellipse and how to graph them.

You're probably very familiar with **circles**. They're completely round and are technically defined as all points that are a given distance, known as the radius, from a defined point called the center. But what if you took a circle and stretched it in opposite directions, either left and right or up and down? You'd end up with a shape that is still round, but is obviously no longer a circle. What you actually have now is an **ellipse**. But what is an ellipse, and how does it work?

An ellipse is defined as the set of all points where the sum of the distances from two fixed points is constant. In other words, there are two fixed points, called **foci** (or the plural of focus). If you trace out all points where the distance from the first focus to a point plus the distance from the second focus to the same point remains constant, you will draw an ellipse. As mentioned, it's essentially a circle where the circle is stretched vertically or horizontally by equal amounts. Here are two examples:

Here is the formula for an ellipse in standard form:

*A*^2, *b*^2, *h*, and *k* are all numbers that determine various characteristics about the ellipse. Note the formula has minus signs in it in front of *h* and *k*.

From the ellipse formula, you can determine the following items about an ellipse:

- Orientation
- Center
- Height
- Width
- Where the foci are located

To determine the orientation, you will compare *a*^2 and *b*^2. If *a*^2 > *b*^2 (or if the bigger number is under the *x*), then it will be horizontal, or wider than it is taller. If *a*^2 < *b*^2, then you have a vertical ellipse whose height is greater than its width.

The center of the ellipse will be a point (*h*, *k*).

To determine the width and height of the ellipse, you'll first find the values of *a* and *b* by taking the square root of *a*^2 and *b*^2. The width will equal 2*a* and the height will equal 2*b*.

Finally, to find the two focus points, or foci, you will need to find the value of *c*. Note that *c* isn't given in the formula, but must be found by first finding the value of *c*^2 by subtracting *a*^2 and *b*^2, the two numbers in the denominators of the formula. You will always subtract the smaller value from the larger value; in other words, *c*^2 should always be a positive number. Then you will take the square root of *c*^2 to get *c*, which tells you how far away from the center, either up and down or left and right, the foci are located.

Now, let's take a look at some example problems, including how to graph an ellipse.

Here is the equation of an ellipse:

*Identify the type of ellipse and then graph the ellipse.*

This ellipse will be horizontal because the number underneath the *x* (25) is larger than the number underneath the *y* (16). The center of the ellipse will be (-2, 0) because *h* = -2 and *k* = 0.

To graph the ellipse, you will first need to find the values of *a* and *b*. The square root of 25, or 5, will be *a*, and the square root of 16, or 4, will be *b*.

Now you will plot the center (-2, 0); then move to the left and right of the center by the value of *a*, or 5, and plot two points. Next, move up and down from the center by the value of *b*, or 4, and plot two more points. You can now connect the four points you just graphed to draw the ellipse.

Let's do another one. The equation of an ellipse is shown here:

First, you should identify *a*, *b*, *c*, *h* and *k*. *h* and *k* will be 3 and 2, respectively, while *a* will be 3, the square root of 9, and *b* will be approximately 4.47, the square root of 20. To find *c*, you should subtract 20 - 9 to get 11 - which is *c*^2 - and then square root it. You will find *c* is approximately 3.32. Now you can describe the ellipse in full.

The center of the ellipse will be (3, 2), and it will be vertical because *b*^2 > *a*^2. It will have a width of 6, 2*a* or 2*3, and a height of 8.94, 2*b* or 2*4.47. Lastly, you can graph the ellipse including its foci.

Start at (3, 2) and move up and down from it by 4.47 units and place two points. Next, go back to the center and move left and right by 3 units and place two more points. You can now connect these 4 points to draw the ellipse. Finally, go back to the center, move up and down by 3.32 units, and draw two points to plot the foci.

To review: if you take a circle and stretch it in opposite directions, either up and down or left and right, you will end up with an oval-shaped figure called an **ellipse**. Every ellipse has a center (*h*, *k*) and two focus points, or foci. When the equation of an ellipse is written in standard form, you can identify its direction, horizontal or vertical; its width, 2*a*; and its height, 2*b*. Finally, you can graph an ellipse by using its center and the values you find for *a*, *b*, and *c* from its equation.

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High School Geometry: Help and Review13 chapters | 162 lessons

- Derive the Equation of a Hyperbola from the Foci 7:07
- Derive the Equation of an Ellipse from the Foci 5:08
- Finding the Equation of a Parabola from the Focus and Directrix 6:16
- Foci and the Definitions of Ellipses and Hyperbolas 6:11
- Practice with the Conic Sections 5:38
- The Focus and Directrix of a Parabola 4:47
- Ellipse: Definition, Equation & Examples 5:41
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