Hyperbolas
Hyperbolas
Hyperbolas vs. Ellipse
Similarities between Hyperbolas and Ellipses:
- The formula is identical, except for the replacement of a "+" with a "-".
- The definition of a is very similar. In a horizontal ellipse, you move horizontally a units from the center to the edges of the ellipse (This defines the major axis). In a horizontal hyperbola, you move horizontally a units from the center to the vertices of the hyperbola (This defines the transverse axis). b defines a different, perpendicular axis.
- The definition of c is identical: the distance from center to focus.
- The biggest difference is that for an ellipse, a is always the biggest of the three variables; for a hyperbola, c is always the biggest. This should be evident from looking at the drawings (the foci are inside an ellipse, outside a hyperbola). However, this difference leads to several other key distinctions.
- For ellipses, c2= a2-b2. For hyperbolas, c2=a2+b2.
- For ellipses, you tell whether it is horizontal or vertical by looking at which denominator is greater, since a must always be bigger than b. For hyperbolas, you tell whether it is horizontal or vertical by looking at which variable has a positive sign, the x2 or the y2. The relative sizes of a and b do not distinguish horizontal from vertical.