# Standard Equation of a Hyperbola

I. Introduction: A hyperbola is a type of conic section defined by the difference of the distances from any point on the curve to two fixed points (foci) being constant. The standard forms of a hyperbola provide a structured way to represent these curves, each with distinct characteristics.

II. General Equation of a Hyperbola: The general equation of a hyperbola with center $\left(h,k\right)$ is given by: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$

III. Standard Forms:

1. Horizontal Form:

• Equation: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$
• Characteristics:
• The major axis is parallel to the x-axis.
• $a$ is the semi-major axis.
• $b$ is the semi-minor axis.
2. Vertical Form:

• Equation: $\frac{\left(y-k{\right)}^{2}}{{b}^{2}}-\frac{\left(x-h{\right)}^{2}}{{a}^{2}}=1$
• Characteristics:
• The major axis is parallel to the y-axis.
• $a$ is the semi-major axis.
• $b$ is the semi-minor axis.

IV. Center and Axes:

1. Center:

• The coordinates $\left(h,k\right)$ represent the center of the hyperbola.
2. Axes:

• The major axis is the longer axis, and the minor axis is perpendicular to it.
• The length of the major axis is $2a$, and the length of the minor axis is $2b$.

V. Asymptotes:

• Asymptotes are lines that the hyperbola approaches but never reaches.
• For a horizontal hyperbola: $y=k±\frac{b}{a}\left(x-h\right)$.
• For a vertical hyperbola: $y=k±\frac{a}{b}\left(x-h\right)$.

VI. Foci and Eccentricity:

• Foci: Points along the transverse axis, at a distance 'c' from the center ($c=\sqrt{{a}^{2}+{b}^{2}}$).

• Eccentricity ($e$): Defined as $e=\frac{c}{a}$.

• $e<1$: Ellipse.
• $e=1$: Parabola.
• $e>1$: Hyperbola.

VII. Special Cases:

1. Standard Form with Center at the Origin:

• $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ (Horizontal)
• $\frac{{y}^{2}}{{b}^{2}}-\frac{{x}^{2}}{{a}^{2}}=1$ (Vertical)
2. Eccentricity and Shape:

• If $e<1$, the hyperbola is an ellipse.
• If $e=1$, the hyperbola is a parabola.
• If $e>1$, the hyperbola is a hyperbola.

VII. Conclusion: Understanding the standard equation of a hyperbola provides a powerful tool for analyzing and graphing these intriguing curves. The key parameters, such as the center, axes, asymptotes, foci, and eccentricity, offer valuable insights into the geometric and algebraic properties of hyperbolas.

Example:

Let's consider the standard equation of a hyperbola and work through an example:

$\frac{\left(x-3{\right)}^{2}}{16}-\frac{\left(y+2{\right)}^{2}}{9}=1$

This equation represents a horizontal hyperbola with its center at $\left(3,-2\right)$, a semi-major axis of length 4 (square root of 16), and a semi-minor axis of length 3 (square root of 9).

1. Identify Key Parameters:

• Center ($\left(h,k\right)$: $\left(3,-2\right)$
• Semi-Major Axis ($a$): $a=4$ (square root of 16)
• Semi-Minor Axis ($b$): $b=3$ (square root of 9)

2. Asymptotes:

• For a horizontal hyperbola, the asymptotes are given by $y=k±\frac{b}{a}\left(x-h\right)$.
• Substituting the values, the asymptotes are $y=-2±\frac{3}{4}\left(x-3\right)$.

3. Foci and Eccentricity:

• Foci: Calculate $c=\sqrt{{a}^{2}+{b}^{2}}=\sqrt{16+9}=5$.
• Foci are located at $\left(3±5,-2\right)$: $\left(8,-2\right)$ and $\left(-2,-2\right)$.
• Eccentricity ($e$): $e=\frac{c}{a}=\frac{5}{4}$.

4. Graphing the Hyperbola:

• Plot the center at $\left(3,-2\right)$.
• Plot the vertices at $\left(3±4,-2\right)$: $\left(7,-2\right)$ and $\left(-1,-2\right)$.
• Plot the foci at $\left(8,-2\right)$ and $\left(-2,-2\right)$.
• Sketch the asymptotes $y=-2±\frac{3}{4}\left(x-3\right)$.
• Draw the hyperbola based on these points and lines.

5. Interpretation:

• The hyperbola is centered at $\left(3,-2\right)$.
• The vertices are at $\left(7,-2\right)$ and $\left(-1,-2\right)$.
• The foci are at $\left(8,-2\right)$ and $\left(-2,-2\right)$.
• The asymptotes guide the behavior of the hyperbola as it extends infinitely.