Hyperbola

Definition

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.

5 Must Know Facts For Your Next Test

1. The standard form of a hyperbola centered at the origin with horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
2. For a hyperbola, $c^2 = a^2 + b^2$, where $c$ is the distance from the center to each focus.
3. 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.
4. Hyperbolas have two foci located along the transverse axis outside each branch.
5. 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.

Related terms

Ellipse: 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.

Parabola: A conic section created when a plane intersects parallel to one generator of the cone, producing an open curve with one branch.

Conic Sections: Curves obtained by intersecting a right circular cone with different angles; includes circles, ellipses, parabolas, and hyperbolas.