Conic Sections: The Hyperbola Dr. Shildneck Fall, 2014

Hyperbolas An HYPERBOLA is a locus of points such that the difference of the distances from two fixed points (the foci) is always the same. The two curves that make up an hyperbola are called branches. The point on each branch closest to the center are the vertices. The branches follow guide lines called asymptotes.

The Equation(s) The positive variable determines the orientation. Positive x – opens horizontally Positive y – opens vertically The number under x indicates horizontal “distance.” The number under y indicates vertical “distance.” The number under the positive indicates the vertices. The other number helps determine the slope of the asymptotes. The distance from the center to focus is c: c2 = a2 + b2

To Graph an Hyperbola Determine the center Determine the horizontal and vertical distances. Mark those distances and make a box. Sketch Asymptotes through the corners of the box. Determine which way it opens. Mark the vertices. Sketch the branches using the vertices and asymptotes.

Example 1: Foci

Writing the Equation of an Hyperbola Determine the center. Determine which way it opens (what comes first)? Determine the distance to the vertices (first denominator). Find the other denominator. (You may be given a focus or the equations of the asymptotes) Plug in all known values to write the equation.

Example 2 Write the equation of the hyperbola centered at the origin with a vertex at (4, 0) and Focus at (7, 0).

Example 3 Write the equation of the hyperbola centered at (2, 5) with a vertex at (2, 8) and asymptote y=(3/2)x + 2.