Once you've been introduced to ellipses, one of your first tasks will usually be to demonstrate that you can extract information about an ellipse from its equation, and also to graph a few ellipses. These will involve some technical processes.

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It will likely be wise to invest a little extra time and effort in practicing and memorizing the steps, so you're in good shape for the test.

- State the center, vertices, foci and eccentricity of the ellipse with general equation 16
*x*^{2}+ 25*y*^{2}= 400, and sketch the ellipse.

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 *x*^{2} = (*x* − 0)^{2} and *y*^{2} = (*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 *a*^{2} is always the larger denominator (and *b*^{2} 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 *a*^{2} = 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 *a*^{2} and *b*^{2}, so:

*a*^{2} − *c*^{2} = *b*^{2}

25 − *c*^{2} = 16

9 = *c*^{2}

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:

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

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

center: (0, 0)

vertices: (−5, 0) and (5, 0)

foci: (−3, 0) and (3, 0)

eccentricity:

You may find it helpful to do the roughing-in of your graph with pencil, rotating the paper as you go around, and then draw your final graph in pen, carefully erasing your rough draft before you hand in your work. And always make sure your graph is neat, that it is large enough to be clear, and that your curves are appropriately rounded. There are no straight, or even straight-ish, line segments on ellipses.

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- State the center, foci, vertices, and co-vertices of the ellipse with equation 25
*x*^{2}+ 4*y*^{2}+ 100*x*− 40*y*+ 100 = 0. Also state the lengths of the two axes.

I first have to rearrange this equation into conics form by completing the square and dividing through to get "=1". Once I've done that, I can read off the information I need from the equation.

25*x*^{2} + 4*y*^{2} + 100*x* − 40*y* = −100

25*x*^{2} + 100*x* + 4*y*^{2} − 40*y* = −100

25(*x*^{2} + 4*x* ) + 4(*y*^{2} − 10*y* ) = −100 + 25( ) + 4( )

25(*x*^{2} + 4*x* + 4) + 4(*y*^{2} − 10*y* + 25) = −100 + 25( 4 ) + 4( 25 )

25(*x* + 2)^{2} + 4(*y* − 5)^{2} = −100 + 100 + 100 = 100

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The larger demoninator is *a*^{2}, and the *y* part of the equation has the larger denominator, so this ellipse will be taller than wide (to parallel the *y*-axis). Also, *a*^{2} = 25 and *b*^{2} = 4, so the equation *b*^{2} + *c*^{2} = *a*^{2} gives me 4 + *c*^{2} = 25, and *c*^{2} must equal 21.

The center is clearly at the point (*h*, *k*) = (−2, 5). The vertices are *a* = 5 units above and below the center, at (−2, 0) and (−2, 10). The co-vertices are *b* = 2 units to either side of the center, at (−4, 5) and (0, 5). The major axis has length 2*a* = 10, and the minor axis has length 2*b* = 4.

The foci are messy: they're units above and below the center.

center: (−2, 5)

vertices: (−2, 0) and (−2, 10)

co-vertices: (−4, 5) and (0, 5)

foci: and

major axis length: 10

minor axis length: 4

As in the example above, you may be given the ellipse's equation in "general" form; that is, with everything multiplied out, so there are no denominators and no parentheticals. But to extract the information from the equation, you need the equation in conics form. To go from the general form to the conics form, you'll need to be sure that you can reliably complete the squares. If you're not comfortable with that process, give yourself some time to do some extra practice before the next test.

URL: https://www.purplemath.com/modules/ellipse2.htm

You can use the Mathway widget below to practice converting general-form ellipse equations to "vertex" or conics form (or skip the widget, and continue to the next page). Try the entered exercise, or type in your own exercise. Then click the button and select "Write in Standard Form" to compare your answer to Mathway's.

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