Rectangular hyperbola
Rectangular Hyperbola
A rectangular hyperbola is a specific type of hyperbola characterized by the property that its asymptotes are perpendicular to each other. This means that the angle between the asymptotes is 90 degrees, which gives the hyperbola a unique set of properties.
Definition
A hyperbola is the set of all points $(x, y)$ in the plane such that the absolute difference of the distances from two fixed points, called foci, is constant. For a rectangular hyperbola, the distance between the foci is equal to $2\sqrt{2}$ times the length of the semi-transverse axis.
Standard Equation
The standard equation of a rectangular hyperbola with its center at the origin and asymptotes aligned with the coordinate axes is given by:
$$ xy = c^2 $$
where $c$ is a constant related to the distance between the foci and the semi-transverse axis.
Properties
Here are some important properties of a rectangular hyperbola:
- The axes of symmetry are the lines $y = x$ and $y = -x$.
- The asymptotes are the coordinate axes themselves.
- The eccentricity $e$ of a rectangular hyperbola is $\sqrt{2}$.
- The product of the perpendicular distances from any point on the hyperbola to the asymptotes is constant and equal to $c^2$.
Table of Differences and Important Points
| Property | Rectangular Hyperbola | General Hyperbola |
|---|---|---|
| Asymptotes | Perpendicular to each other | Not necessarily perpendicular |
| Eccentricity | $e = \sqrt{2}$ | $e > 1$, not fixed |
| Equation | $xy = c^2$ | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ |
| Axes of Symmetry | Lines $y = x$ and $y = -x$ | Not necessarily along $y = x$ or $y = -x$ |
| Distance between Foci | $2\sqrt{2}a$ | $2\sqrt{a^2 + b^2}$ |
Formulas
Asymptotes
For the rectangular hyperbola $xy = c^2$, the asymptotes are the coordinate axes, given by:
$$ x = 0 \quad \text{and} \quad y = 0 $$
Foci
The foci of a rectangular hyperbola are given by:
$$ F_1 = (c\sqrt{2}, c\sqrt{2}) \quad \text{and} \quad F_2 = (-c\sqrt{2}, -c\sqrt{2}) $$
Eccentricity
The eccentricity of a rectangular hyperbola is always:
$$ e = \sqrt{2} $$
Directrices
The directrices of a rectangular hyperbola are given by:
$$ x = \pm \frac{c}{\sqrt{e}} \quad \text{and} \quad y = \pm \frac{c}{\sqrt{e}} $$
Examples
Example 1: Finding the Asymptotes
Given the rectangular hyperbola $xy = 16$, find the asymptotes.
Solution:
The asymptotes are the coordinate axes, so they are:
$$ x = 0 \quad \text{(y-axis)} \quad \text{and} \quad y = 0 \quad \text{(x-axis)} $$
Example 2: Finding the Foci
Given the rectangular hyperbola $xy = 9$, find the foci.
Solution:
First, we find $c$ such that $c^2 = 9$, which gives $c = 3$. The foci are then:
$$ F_1 = (3\sqrt{2}, 3\sqrt{2}) \quad \text{and} \quad F_2 = (-3\sqrt{2}, -3\sqrt{2}) $$
Example 3: Eccentricity
Given any rectangular hyperbola, such as $xy = 25$, find the eccentricity.
Solution:
The eccentricity of a rectangular hyperbola is always $\sqrt{2}$, regardless of the value of $c^2$.
$$ e = \sqrt{2} $$
In summary, the rectangular hyperbola is a special case of the hyperbola with unique properties that distinguish it from the general hyperbola. Its perpendicular asymptotes and constant eccentricity make it an interesting subject of study in mathematics.