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Question 392638: describe the similarities and differences between hyperbolas and ellipses
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! describe the similarities and differences between hyperbolas and ellipses
standard form of the ellipse: (x-h)^2/a^2+(y-k)^2/b^2=1 (a always greater than b)
or (y-k)^2/a^2+(x-h)^2/b^2=1 (a always greater than b)
standard form of the hyperbola: (x-h)^2/a^2-(y-k)^2/b^2=1
a greater than b, less than b or equal to b
or( y-k)^2/a^2-(x-h)^2/b^2=1
a greater than b, less than b or equal to b
First, you might notice the standard forms for ellipses and hyperbolas are almost the same except for a negative sign in the hyperbola, but they look very different as seen by the graphs above.
Both curves have vertices, foci, and a center. The (h,k) in both forms are the (x,y)coordinates of the centers. Both curves shown above have centers at the origin (0,0)
I have shown two forms for each of the curves. Let me explain the ellipse first. The first form listed shows a^2 as the denominator for the x^2 term. In an ellipse, a is always greater than b. If a, the larger
of the two, is under the x^2 term, the major axis is horizontal. If a is under the y^2 term, then the major axis of the ellipse is vertical. The ellipse shown above is this latter type.
In case of the hyperbola, a could be greater than b, less than b, or equal to b. If the x^2 term is listed first, its transverse axis would be horizontal. Conversely, if the y^2 term is listed first, the transverse axis would be vertical. The hyperbola shown above is the latter type.
In an ellipse the distance from vertex to center is equal to a. The length of the minor axis is equal to b. The foci, usually named c, is equal to the sqrt(a^2-b^2). Note that the distance from center to vertex in an ellipse is longer than the distance to the foci. It is the other way around in a hyperbola, the foci distance from the center is longer than the vertex distance. It follows that the foci distance ,c, in a hyperbola is equal to the sqrt(a^2+b^2)
Hyperbolas also have asymptotes ( not shown above) that ellipses don't have. Their slopes are +-a/b for the vertical transverse axis, and +-b/a for the horizontal transverse axis and they all go thru the center.
There is probably more to it than I have described, but this should give you a good idea of some of the main differences and similarities between ellipses and hyperbolas.
Hope this helps!
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