Hyperbola: Rectangular Hyperbola and Asymptotes A hyperbola is a shape formed by the distance between two points called the vertices . The foci ar...
A hyperbola is a shape formed by the distance between two points called the vertices. The foci are points along the major axis at a fixed distance from each vertex. The major axis is a line segment joining the vertices, and the minor axis is a line segment connecting the foci.
A rectangular hyperbola is a special type of hyperbola where the distance between the vertices is equal to the distance between the foci. This means that the foci are on the same line as the vertices.
The asymptotes of a hyperbola are two lines that approach the hyperbola as it approaches infinity. The upper asymptote is the line that is above the hyperbola, and the lower asymptote is the line that is below the hyperbola.
Here's a table summarizing the key characteristics of a hyperbola:
| Feature | Hyperbola with Center at (a, b) |
|---|---|
| Vertices | (a, -b) and (a, b) |
| Major axis | x = a |
| Minor axis | x = -a |
| Foci | F(a, 0) and F(-a, 0) |
| Asymptotes | y = x and y = -x |
Examples:
The hyperbola with center at (3, 4), with vertices at (1, 7) and (7, 7), is a rectangular hyperbola.
The hyperbola with center at (5, 1), with focus at (3, 5), has vertical asymptotes.
The hyperbola with center at (2, 6), with major axis length 10, has two branches with equation x = ±2