HYPERBOLAHYPERBOLA

DefinitionDefinition: :

A A hyperbolahyperbola is all points found by keeping the difference of the is all points found by keeping the difference of the distances from two points (each of which is called a distances from two points (each of which is called a focusfocus of the of the hyperbola) constant. The midpoint of the segment (the hyperbola) constant. The midpoint of the segment (the transverse transverse axisaxis) connecting the foci is the ) connecting the foci is the centercenter of the hyperbola. Am of the hyperbola. Am hyperbola can be formed by slicing a right circular cone with a plane hyperbola can be formed by slicing a right circular cone with a plane traveling parallel to the vertical axis of the cone. This effect can be traveling parallel to the vertical axis of the cone. This effect can be seen in the following seen in the following videovideo and screen captures. and screen captures.

Part I: Hyperbolas center at the Part I: Hyperbolas center at the origin.origin.

Example : In the first example the constant distance mentioned above will be 6, one focus will be at the point (0, 5) and the other will be at the point (0, -5).The graph of a hyperbola with these foci and center at the origin is shown below.

An equation of this hyperbola can be found by using the distance formula. We calculate the distance from the point on the ellipse (x, y) to the two foci, (0, 5) and (0, -5). This total distance is 6 in this example:

Note that 6 is the total distance from vertex to vertex through the center of this hyperbola, shown by the dark red line on the graph. This is called the transverse axis.

After eliminating radicals and simplifying we have

If we let a = 4 and b = 3, this equation can be written as

. The important features are: •b = 3, the distance from the center to the vertices of the hyperbola in the vertical direction (up and down from the center). This is called the transverse axis. a = 4, and the two dotted lines in the graph are given by the equations

is the distance from the center to each focus. Each focus is found on the transverse axis. The two dotted lines on the graph are asymptotes because the two branches of the hyperbola approach but never reach these lines. Sketching them first provides a good way to sketch the graph of a hyperbola.