A hyperbola is a type of conic section formed by intersecting a double cone with a plane such that the angle between the plane and the cone's axis is less than that between the plane and one of the cone's generators. It consists of two symmetric open curves called branches.
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The standard form of a hyperbola centered at the origin with horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
For a hyperbola, $c^2 = a^2 + b^2$, where $c$ is the distance from the center to each focus.
The asymptotes of a hyperbola in standard form are given by $y = \pm \frac{b}{a}x$ for a horizontal transverse axis and $y = \pm \frac{a}{b}x$ for a vertical transverse axis.
Hyperbolas have two foci located along the transverse axis outside each branch.
In polar coordinates, hyperbolas can be represented using equations like $r = \frac{ed}{1 + e\cos(\theta)}$, where e > 1.
Review Questions
What is the relationship between $a$, $b$, and $c$ in a hyperbola?
How do you find the equations of the asymptotes for a hyperbola?
Describe how to represent a hyperbola in polar coordinates.
A conic section formed by cutting through both nappes of a cone with an oblique plane such that it does not intersect them, resulting in an oval shape.