The standard form of a hyperbola describes its orientation and position on a coordinate plane, with two primary forms depending on whether it opens sideways (horizontally) or up and down (vertically).
The exact standard forms of a hyperbola are:
- (x - h)² / a² - (y - k)² / b² = 1 for hyperbolas that open sideways.
- (y - k)² / a² - (x - h)² / b² = 1 for hyperbolas that open up and down.
In both forms, the center of the hyperbola is located at the coordinates (h, k).
Understanding the Standard Forms
The standard equations provide crucial information about the hyperbola's characteristics, including its center, orientation, and the lengths of its semi-transverse and semi-conjugate axes.
1. Hyperbola Opening Sideways (Horizontal Transverse Axis)
When the x term comes first in the equation, the hyperbola opens horizontally, meaning its branches extend left and right.
Formula: (x - h)² / a² - (y - k)² / b² = 1
- Center:
(h, k) - Orientation: Opens left and right. The transverse axis is parallel to the x-axis.
a: The distance from the center to each vertex along the transverse axis. The vertices are(h ± a, k).b: Half the length of the conjugate axis. Used to find the co-vertices and to construct the fundamental rectangle, which helps in drawing the asymptotes. The co-vertices are(h, k ± b).
Example:
Consider the equation (x - 2)² / 9 - (y + 1)² / 16 = 1.
- Here,
h = 2,k = -1,a² = 9(soa = 3), andb² = 16(sob = 4). - The center is
(2, -1). - The hyperbola opens sideways.
2. Hyperbola Opening Up and Down (Vertical Transverse Axis)
When the y term comes first in the equation, the hyperbola opens vertically, meaning its branches extend upwards and downwards.
Formula: (y - k)² / a² - (x - h)² / b² = 1
- Center:
(h, k) - Orientation: Opens up and down. The transverse axis is parallel to the y-axis.
a: The distance from the center to each vertex along the transverse axis. The vertices are(h, k ± a).b: Half the length of the conjugate axis. The co-vertices are(h ± b, k).
Example:
Consider the equation (y - 5)² / 25 - (x - 3)² / 4 = 1.
- Here,
h = 3,k = 5,a² = 25(soa = 5), andb² = 4(sob = 2). - The center is
(3, 5). - The hyperbola opens up and down.
Summary Table of Standard Forms
| Feature | Horizontal Hyperbola | Vertical Hyperbola |
|---|---|---|
| Equation | (x - h)² / a² - (y - k)² / b² = 1 |
(y - k)² / a² - (x - h)² / b² = 1 |
| Center | (h, k) |
(h, k) |
| Orientation | Opens left and right | Opens up and down |
| Vertices | (h ± a, k) |
(h, k ± a) |
| Co-vertices | (h, k ± b) |
(h ± b, k) |
| Dominant Term | x term is positive |
y term is positive |
Understanding these standard forms is fundamental for graphing hyperbolas and solving problems related to their properties, such as finding foci or asymptotes.
