Ellipses: Finding Information From The Equation - Purplemath

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To be able to read any information from this equation, I'll need to rearrange it to get the variable terms grouped together, so I get an equation that is "=1". First, I'll divide through by the 400:

Since x2 = (x − 0)2 and y2 = (y − 0)2, the equation above can be more-helpfully restated as:

Then the center is at (h, k) = (0, 0).

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I know that the a2 is always the larger denominator (and b2 is the smaller denominator), and this larger denominator is under the variable that parallels the longer direction of the ellipse. Since 25 is larger than 16, then a2 = 25, a = 5, and this ellipse is wider (paralleling the x-axis) than it is tall. The value of a also tells me that the vertices are five units to either side of the center, so they're at (−5, 0) and (5, 0).

To find the foci, I need to find the value of c. From the equation, I already have a2 and b2, so:

a2 − c2 = b2 25 − c2 = 16 9 = c2

Then the value of c is 3, and the foci are three units to either side of the center, at the points (−3, 0) and (3, 0).

Also, the value of the eccentricity e is .

To sketch the ellipse, I first draw the dots for the center and for the endpoints of each axis:

axis system with important graphing points drawn in

Then I rough in a curvy line, rotating my paper as I go and eye-balling my curve for smoothness...

outline of ellipse roughed in

...and then I draw my "answer" as a heavier solid line.

final graph of ellipse

center: (0, 0) vertices: (−5, 0) and (5, 0) foci: (−3, 0) and (3, 0) eccentricity:

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