lesson on hyperbola

General Form of Conic Sections

The general equation for all conic sections is $Ax^2 + Cy^2 + Dx + Ey + F = 0$.

Circle Identification

A conic section is a circle if $A = C$ in the general form.

Ellipse Identification

A conic section is an ellipse if $A \neq C$ and $AC > 0$ (same sign).

Hyperbola Identification

A conic section is a hyperbola if $AC < 0$ (opposite signs).

Parabola Identification

A conic section is a parabola if either $A = 0$ or $C = 0$, but not both.

Definition of Hyperbola

A hyperbola is the set of all points in a plane whose distance from two fixed points (foci) has a constant difference.

Distance Difference in Hyperbola

The constant difference in a hyperbola equals $2a$.

Standard Form of Horizontal Hyperbola

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$; opens left and right.

Standard Form of Vertical Hyperbola

$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$; opens up and down.

Center of Hyperbola

The center of a hyperbola is denoted as $(h, k)$ where $h$ and $k$ are the coordinates.

Vertices of Hyperbola

Vertices are the two points where the hyperbola intersects the transverse axis, located $a$ units from the center.

Foci of Hyperbola

Foci are the two fixed points that define the hyperbola, located $c$ units from the center.

Conjugate Axis of Hyperbola

The conjugate axis is the line segment perpendicular to the transverse axis, passing through the center.

Asymptotes of Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches.

Pythagorean Relationship for Hyperbola

The relationship between $a$, $b$, and $c$ in hyperbolas is given by $a^2 + b^2 = c^2$.

Standard Form of Ellipse

For horizontal orientation, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$; for vertical, switch the terms.

Key Difference Between Hyperbola and Ellipse

Ellipses use addition while hyperbolas use subtraction in their equations.

Circle Definition

A circle is a set of all points equidistant from a fixed center point, with distance equal to radius $r$.

Standard Form of Circle

The standard form of a circle is $(x-h)^2 + (y-k)^2 = r^2$.

Parabola Definition

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

Steps to Convert General Form to Standard Form

Identify conic type, rearrange terms, complete the square, factor coefficients, and divide through by the constant.

Completing the Square for Hyperbolas

Factor out the coefficient to make the squared term equal 1, then complete the square.

Common Graphing Mistakes

Confusing signs of $h$ and $k$, reading coordinates left-to-right, and misapplying orientation-specific formulas.

Relationship Between $a$, $b$, and $c$ in Ellipses

For ellipses, the relationship is $a^2 = b^2 + c^2$.

Graphing Hyperbola Steps

Identify orientation, plot center, plot vertices, find conjugate axis endpoints, draw auxiliary rectangle, plot asymptotes, sketch the curves.