Hyperbola Overview and Properties

Overview

This lesson introduces the hyperbola, its standard equations, parts, and key properties, including axes, vertices, foci, and asymptotes.

The standard form is ( x^2/a^2 - y^2/b^2 = 1 ) for a horizontal transverse axis.
The form ( y^2/a^2 - x^2/b^2 = 1 ) is for a vertical transverse axis.
Centered at ((h, k)), the equation is ((x-h)^2/a^2 - (y-k)^2/b^2 = 1) for horizontal.
For vertical, it is ((y-k)^2/a^2 - (x-h)^2/b^2 = 1).

Hyperbolas are sets of points where the absolute difference of distances to two fixed points (foci) is constant.
Each hyperbola has two branches, which can open left/right or up/down.
The transverse axis connects the vertices of the hyperbola.
The center is the midpoint between the vertices.
Each branch has a vertex at its "turning" point.
The distance from the center to a focus is ( c = \sqrt{a^2 + b^2} ).
The conjugate axis is perpendicular to the transverse axis.

The fundamental rectangle is formed by connecting the vertices and endpoints of axes.
Asymptotes of ( x^2/a^2 - y^2/b^2 = 1 ) are lines through the center with slopes ( \pm b/a ).
The equations for asymptotes are ( y = (b/a)x ) and ( y = -(b/a)x ).
Asymptotes pass through the diagonals of the rectangle and guide the branches of the hyperbola.

Hyperbola — The set of points where the absolute difference of distances to two foci is constant.
Transverse Axis — The axis joining the vertices, the major axis of the hyperbola.
Conjugate Axis — The axis perpendicular to the transverse axis.
Vertex (of Hyperbola) — The point where each branch turns, located on the transverse axis.
Center (of Hyperbola) — The midpoint between the vertices or intersection of axes.
Foci (plural of Focus) — Two fixed points defining the hyperbola.
Asymptote — A line that the hyperbola approaches but never touches.