Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Ellipses and Hyperbolas
Many of us are aware that the earth orbits the sun by traveling around it. However, what some people might not be aware of, is that the path the earth takes around the sun is the shape of a mathematical curve called an ellipse.
Wow! That's so cool! An ellipse is a mathematical shape that looks somewhat like a circle that has been squashed down a bit. Ellipses were first discovered by the ancient Greeks during their study of conic sections, where conic sections are curves that can be obtained by slicing a right cone at different angles. The ellipse is found by slicing a right cone as shown.
Another conic section studied by the Greeks was the hyperbola. A hyperbola is a conic section that can be obtained by slicing a right cone as shown on screen now.
When we slice the cone in this way, the cross section is a hyperbola and looks somewhat like two U's with their bottoms facing each other.
So far, this is pretty neat! Who would have though that slicing a cone could be so interesting! Let's take a look at a characteristic of both the ellipse and the hyperbola that will allow us to examine each curve more technically.
Foci of an Ellipse and Hyperbola
Both an ellipse and a hyperbola have certain points called foci. Let's take a look at these points in each of the curves.
First, let's talk about the foci of an ellipse. The foci of an ellipse are two points, F and G, such that the distance from F to any point P, on the ellipse, to G is always the same.
This information allows us to give a more technical definition of an ellipse. That is, given two points, F and G, an ellipse is the set of points P, such that FP + PG is constant, and we call the points, F and G, the ellipse's foci.
As we said, it turns out that hyperbolas also have foci. To understand the foci of a hyperbola, it's best to define the hyperbola in terms of its foci. Doing this will also give us the technical definition of a hyperbola, so let's give it a go!
For two given points, F and G called the foci, a hyperbola is the set of points, P, such that the difference between the distances, FP and GP, is constant. That is, if F and G are the foci of a hyperbola, then for any point, P, on the hyperbola, the absolute value of FP - GP is constant.
Hmmm…that's a lot of information. I don't know about you, but I prefer to just think of an ellipse as a squashed circle and a hyperbola as two U's with their bottoms facing each other. Nonetheless, knowing what the foci is for each of these curves, along with their technical definitions, is very important when studying these concepts.
Let's take a look at a couple of examples to further our understanding of ellipses and hyperbolas.
First, consider the ellipse shown with foci (-4, 0) and (4, 0). The image shows this ellipse with two points on the ellipse labeled.
Notice that the distances from each of the foci to the points on the ellipse are as follows:
FP1 = 17 / 5
GP1 = 33 / 5
FP2 = 29 / 5
GP2 = 21 / 5
By the technical definition of an ellipse, it should be the case that:
FP1 + GP1 = FP2 + GP2.
Well, this is pretty simple to verify!
FP1 + GP1 = (17 / 5) + (33 / 5) = 50 / 5 = 10
FP2 + GP2 = (29 / 5) + (21 / 5) = 50 / 5 = 10
Look at that! It turns out exactly as it should, according to the definition of an ellipse!
Let's look at a similar example with a hyperbola. The image you are looking at shows a hyperbola with its foci and two points labeled.
We see that the distances between the foci and the labeled points are as follows:
FP1 = 4
GP1 = 8
FP2 = 14
GP2 = 10
By the technical definition of the hyperbola, it should be the case that:
|FP1 - GP1| = |FP2 - GP2|
Once again, let's go ahead and verify this is true:
|FP1 - GP1| = |4 - 8| = 4
|FP2 - GP2| = |14 - 10| = 4
Excellent! Once again, everything checks out, and these examples make those technical definitions a little clearer.
Let's review. An ellipse is a mathematical shape that looks somewhat like a circle that has been squashed down a bit, and a hyperbola is a conic section that can be obtained by slicing a right cone. Put simply, an ellipse and a hyperbola are conic sections, or curves that can be obtained by slicing a right cone at different angles, with the shapes of a squashed circle and two U's with their bottoms facing each other, respectively. Both an ellipse and a hyperbola have points called foci, and we use these points to define both of these curves technically as follows:
- Given two points, F and G, an ellipse is the set of points, P, such that FP + PG is constant, and we call the points, F and G, the ellipse's foci.
- For two given points, F and G, called the foci, a hyperbola is the set of points, P, such that the difference between the distances, FP and GP is constant.
Though these technical definitions may sound a bit daunting, by working with these curves and their foci, they become clearer, and these curves are better understood.
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