High School: Geometry
Expressing Geometric Properties with Equations HSG-GPE.A.3
3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
For many students, solving problems in Geometryland can sometimes feel like a business trip to Algebraland. (It isn't a leisurely vacation, that's for sure!) Working with the equations of ellipses and hyperbolas requires fluency in two dialects of Mathese: both Geospeak and Algetongue.
Focus, vertex, asymptote, transverse, conjugate, standard position, Pythagorean theorem—all important geometric terms that students need to know. On the other hand, square, simplify, solve for x, and complete the square are crucial algebraic skills they need to be able to perform.
If their trip to Geometryland proves successful, they should know the equations that describe both ellipses (which look like elongated circles) and hyperbolas (which look like a pair of parabolas). For ellipses or hyperbolas with center (h, k), a major axis of a, and a minor axis of b, the formulas are:
Students should know that when the major axes are vertical rather than horizontal, the a2 and b2 values are switched. Whichever axis is elongated gets the larger a underneath it.
Chances are students will also want to know the distance formula. After all, the standard itself features the word "distances." Lucky for them, the distance formula is still the same as it always was.
Students should understand that while parabolas have a single focus, ellipses and hyperbolas have two of them. (The plural of focus is "foci" so don't let that trip them up.) The distance from one focus to any point on the ellipse and back to the other focus will always be constant. The same applies to hyperbolas, only instead of adding the distances, we subtract them.
Given a set of two foci, students should be able to find the equation of an ellipse. They can do this by setting up the equation in the form of two distances calculations (the distance from one focus to (x, y) and the other focus to that same point) that equal a constant, or by finding the values of h, k, a, and b.
It also helps to imagine a right triangle inside the ellipse with vertices at the center, one of the foci, and at the vertical vertex. The length of the hypotenuse is half the constant distance c (the other half is the hypotenuse of the triangle on the other side), the length of the base helps find the location of the foci, and the length of the vertical leg is b (the same b as in the equations above). Students can also find a by dividing the constant distance by 2.
You might consider representing the constant distance as a string loosely connected to two foci. Using the string to guide your marker, draw the ellipse and explain the relationship between the values and distances. Bringing in triangles will help mathematically cement these relationships. Students should familiarize themselves with all the intricacies of Ellipseville so that they can actually get around (and around and around).
The itinerary for Hyperbola City should look about the same. Students should be able to perform essentially the same calculations, finding the equation when given the foci. The only difference is that instead of a sum of two distances, we're talking about a difference.
If students are struggling, they might benefit from:
- reviewing the algebra used to simplify big equations, especially completing the square
- considering how the Pythagorean Theorem is applied here
- focusing on when and where to use addition and subtraction
- referring to a labeled drawing of the shape in question