Introduction to Ellipses
Most ellipses look like flattened circles. Usually one diameter is longer than the other. In Figure 12.11, the horizontal diameter is longer than the vertical diameter. In Figure 12.12 the vertical diameter is longer than the horizontal diameter. The longer diameter is the major axis , and the shorter diameter is the minor axis . An ellipse has seven important points—the center, two endpoints of the major axis (the vertices), two endpoints of the minor axis, and two points along the major axis called the foci (plural for focus ). When the equation of an ellipse is in the form
we can find these points without much trouble.
Sketching Ellipses Graphs
If all we want to do is to sketch the graph, all we really need to do is to plot the endpoints of the diameters and draw a rounded curve connecting them. For example, if we want to sketch the graph of , a = 3, b = 2, h = −1, and k = 1. According to Figure 12.12, the diameters have coordinates (−1 −2, 1) = (−3, 1), (−1 + 2, 1) = (1, 1), (−1, 1 + 3) = (−1, 4), and (−1, 1 − 3) = (−1, −2). (See Figure 12.13.)
DEFINITION: An ellipse is the set of all points whose distances to two fixed points (the foci) is constant.
For example, the foci for . If we take any point on this ellipse and calculate its distance to (−4, 0) and to (4, 0) and add these numbers, the sum will be 10. Two points on this ellipse are (0, 3) and .
Distance from (0, 3) to (−4, 0) + Distance from (0, 3) to (4, 0)
Distance from to (−4, 0) + Distance from to (4, 0)
Finding the Center, Foci, and Vertices of Ellipses Examples
In the next set of problems, we will be given an equation for an ellipse. From the equation, we can find h, k, a , and b . With these numbers and the information in Figures 12.11 or 12.12 we can find the center, foci, and vertices.
Find the center, foci, and vertices for the ellipse.
From the equation, we see that h = 3, k = −5, a 2 and b 2 are 4 2 and 5 2 , but which is a and which is b ? a needs to be the larger number, so a = 5 and b = 4. This makes . We need to use the information in Figure 12.12 because the larger denominator is under ( y − k ) 2 .
Center: ( h , k ) = (3, −5)
Foci: ( h , k − c ) = (3, −5 − 3) = (3, −8) and ( h , k + c ) = (3, −5 + 3) = (3, -2)
Vertices: ( h , k − a ) = (3, −5−5) = (3, −10) and ( h , k + a ) = (3, −5 + 5) = (3, 0)
To make it easier to find h, k, a , and b , we will rewrite the equation.
Now we can see that h = 0, k = 2, a = 4, b = 1, . Because the larger denominator is under ( x − 0) 2 , we need to use the information in Figure 12.11 .
Center: ( h , k ) = (0, 2)
Vertices: ( h − a , k ) = (0 − 4, 2) = (−4, 2) and ( h + a , k ) = (0 + 4, 2) = (4, 2)
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