The Hyperbola

The Hyperbola

A hyperbola is the locus of a point moving such that the difference of its distances from two fixed points (called foci) is constant. This constant difference is equal to the length of the transverse axis ($2a$).

Key Components

• Center ($C$): The midpoint of the transverse axis.
• Vertices ($V$): The endpoints of the transverse axis.
• Transverse Axis: The axis containing the foci and vertices. Its length is $2a$.
• Conjugate Axis: The axis perpendicular to the transverse axis at the center. Its length is $2b$.
• Foci ($F$): Two fixed points on the transverse axis.
• Asymptotes: Two lines that the hyperbola approaches as it extends to infinity.
• Relationship: $c^2 = a^2 + b^2$ (where $c$ is the distance from center to focus).

Standard Equations

Let $(h, k)$ be the center.

Horizontal Hyperbola (Transverse Axis is Horizontal)

Horizontal Hyperbola Equation

$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

• Vertices: $(h \pm a, k)$
• Foci: $(h \pm c, k)$ where $c = \sqrt{a^2 + b^2}$
• Asymptotes: $y - k = \pm \frac{b}{a}(x - h)$

Vertical Hyperbola (Transverse Axis is Vertical)

Vertical Hyperbola Equation

$\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

• Vertices: $(h, k \pm a)$
• Foci: $(h, k \pm c)$ where $c = \sqrt{a^2 + b^2}$
• Asymptotes: $y - k = \pm \frac{a}{b}(x - h)$

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Solved Problems