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Ellipses Help (page 2)

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By and McGraw-Hill Professional
Updated on Oct 4, 2011

Matching the Equation of an Ellipse to its Graph Examples

Now that we can find this important information from an equation of an ellipse, we are ready to match graphs of ellipses to their equations.

Examples

Match the equations with the graphs in Figures 12.14-12.16.

Fig. 12.14

Fig. 12.15

Fig. 12.16

• The larger denominator is under y 2 , so we need to use the information in Figure 12.12 . Because a = 3, we need to look for a graph with vertices (0, 3) and (0, −3). This graph is in Figure 12.15.

• Fig. 12.15

• The larger denominator is under x 2 , so we need to use the information in Figure 12.11 . Because a = 3, the vertices are (−3, 0) and (3, 0). This graph is in Figure 12.16.

• Fig. 12.16

• The larger denominator is under y 2 , so we need to use the information in Figure 12.12 . Because a = 2, the vertices are (0, 2) and (0, −2). This graph is in Figure 12.14.

• Fig. 12.14

Finding the Equation of Ellipses

With as little as three points, we can find an equation of an ellipse. Using the formulas in Figures 12.11 and 12.12 and some algebra, we can find h, k, a , and b .

Fig. 12.11

Fig. 12.12

Examples

Find an equation of the ellipse.

• The vertices are (−4, 2) and (6, 2), and (1, 5) is a point on the graph. The y -coordinates are the same, so the major axis (the larger diameter) is horizontal, which means we need to use the information in Figure 12.11. The vertices are ( ha, k ) and ( h + a, k ). This means that ha = −4 and h + a = 6, and k = 2.

So far we know that

Because (1, 5) is on the graph, . Solving this equation for b , we find that b = 3. The equation is

.

• The foci are (−4, −10) and (−4, 14) and (−4, 15) is a vertex.
• The x -values of foci are the same, so the major axis is vertical. This tells us that we need to use the information in Figure 12.12 .

( h , kc ) = (−4, −10) and ( h , k + c ) = (−4, 14), so h = −4, kc = −10 and k + c = 14.

Because (−4, 15) is a vertex, k + a = 15, so 2 + a = 15 and a = 13. All we need to finish is to find b . Let a = 13 and c = 12 in : . Solving this for b , we have b = 5. The equation is

Finding the Eccentricity of Ellipses

The eccentricity of an ellipse is a number that measures how flat it is. The formula is . This number ranges between 0 and 1. The closer to 1 the eccentricity of an ellipse is, the flatter it is. If , then the ellipse is a circle. In a circle, the center and foci are all the same point, and a and b are the same number. For example, is a circle with center (5, 4) and radius . Usually we see equations of circles in the form ( xh ) 2 + ( yk ) 2 = r 2 .

Examples

Find the ellipse’s eccentricity.

a = 13, b = 12, , This ellipse is more rounded than the first because e is closer to 0.

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