Honors Precalculus 6.4: Hyperbolas Study Guide

Learning Objectives

• Reduce the general equation of the hyperbola to standard form and be able to graph the hyperbola.

• Find the equation of the hyperbola satisfying specific given conditions.

Definition of a Hyperbola

• A Hyperbola is defined as the set of all points such that the difference of its distance from two fixed points is constant.

Standard Equations of the Hyperbola

1. Horizontal Hyperbolas

• Orientation: The transverse axis is parallel to the x-axis.

• Equations:

  • Center at $(0,0)$:     $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

  • Center at $(h, k)$:     $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Equation of the Asymptotes:   $y - k = \pm \frac{b}{a}(x - h)$

2. Vertical Hyperbolas

• Orientation: The transverse axis is parallel to the y-axis.

• Equations:

  • Center at $(0,0)$:     $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

  • Center at $(h, k)$:     $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

• Equation of the Asymptotes:   $y - k = \pm \frac{a}{b}(x - h)$

General Equations of the Hyperbola

• Formula: $Ax^2 - By^2 + Dx + Ey + F = 0$

• Characteristics: The equation is quadratic in two variables where the coefficients of the quadratic terms ($x^2$ and $y^2$) are not equal and have opposite signs.

Properties of the Hyperbola

• Center: Denoted as $(h, k)$.

• Axes of Symmetry:

  • Transverse Axis: The segment containing the vertices; total length = $2a$.

  • Conjugate Axis: The segment perpendicular to the transverse axis; total length = $2b$.

  • Comparison: The transverse axis can be longer, shorter, or equal in length to the conjugate axis.

• Vertices: A hyperbola has four vertices:

  • Ends of Transverse Axis: $V_1, V_2$

  • Ends of Conjugate Axis: $V_3, V_4$

• Foci: Denoted as $F_1, F_2$

  • The distance between the foci = $2c$.

• Lateral Recta (Latus Rectum):

  • The distance from the focus to one end of the latus rectum = $\frac{b^2}{a}$.

  • The total length of the latus rectum = $\frac{2b^2}{a}$.

• Directrices:

  • The distance from the center to a directrix = $\frac{a^2}{c}$.

  • The total distance between the two directrices = $\frac{2a^2}{c}$.

• Slant Asymptotes: These are the diagonals of the rectangle formed by the vertices. Specifically, the vertices of the hyperbola serve as the midpoints of the sides of this rectangle.

• Pythagorean Relationship:

  • Formula: $c^2 = a^2 + b^2$

  • Note that $c$ is the longest segment and represents the distance from the center to the focus.

Eccentricity

• Definition: Eccentricity is a parameter associated with conic sections that measures how much a conic section varies from being a circle.

• Eccentricity Formula: $e = \frac{c}{a}$

• Values for Different Conic Sections:

  • Ellipse: $0 < e < 1$

  • Circle: $e = 0$

  • Parabola: $e = 1$

  • Hyperbola: $e > 1$

Procedures for Finding the Equation

To determine the equation of a hyperbola, three main components are required:

1. Center: The $(h, k)$ coordinates.

2. The value of $a$ and $b$:

   • The distance from the center to one end of the transverse axis is $a$.

   • The distance from the center to one end of the conjugate axis is $b$.

3. Orientation:

   • Vertical: The transverse axis is parallel to the y-axis.

   • Horizontal: The transverse axis is parallel to the x-axis.

Exercises

Exercise A: Sketch the graph

1. $4(x + 2)^2 - 5(y - 1)^2 = 20$

2. $y^2/8 - x^2/3 = 1$ (expressed in some contexts as $3y^2 - 8x^2 + 8x + 3 = 0$ or simplified variations)

3. $3x^2 - 4y^2 + 12x - 16y - 32 = 0$

Exercise B: Find the equation based on properties

1. Vertices: $(0, \pm 2)$, Foci: $(0, \pm 4)$.

2. Vertices: $(\pm 1, 0)$, Asymptotes: $y = \pm 5x$.

3. Foci: $(0, \pm 3\sqrt{17})$, Asymptotes: $y = \pm 4x$.

4. Center at the origin, transverse axis on the y-axis, eccentricity: $\frac{\sqrt{53}}{2}$, distance between foci: $2$.