Understanding Hyperbolas: Equations, Characteristics, and Examples

Posted on Aug 23, 2024 in Physics

Hyperbola as a Geometric Locus

The hyperbola is defined as the geometric locus of all points in a plane where the difference of distances from two fixed points (called foci) is constant.

The equation of a hyperbola is similar to that of an ellipse, but instead of the sum of distances, it involves the difference of distances.

Hyperbola Diagram

Ordinary Form of the Hyperbola Equation

Hyperbola with Horizontal Focal Axis and Center at the Origin (0, 0)

Horizontal Hyperbola Diagram

The equation for this hyperbola is: x2/a2 – y2/b2 = 1

Characteristics:

  • Symmetric with respect to the x and y axes.
  • Foci: F (±c, 0)
  • Vertices: V (±a, 0)
  • Minor Axis Endpoints: B (0, ±b)
  • Center: Origin (0, 0)
  • Transverse Axis Length: 2a
  • Conjugate Axis Length: 2b
  • Focal Axis Length: 2c
  • Eccentricity: e = c/a (where c2 = a2 + b2)
  • Latus Rectum: 2b2/a
  • Directrix: x = ±a2/c
  • Asymptotes: y = ±(b/a)x

Hyperbola with Vertical Focal Axis and Center at the Origin (0, 0)

Vertical Hyperbola Diagram

The equation for this hyperbola is: y2/a2 – x2/b2 = 1

Characteristics:

  • Foci: F (0, ±c)
  • Vertices: V (0, ±a)
  • Minor Axis Endpoints: B (±b, 0)
  • Center: Origin (0, 0)
  • Transverse Axis Length: 2a
  • Conjugate Axis Length: 2b
  • Focal Axis Length: 2c
  • Eccentricity: e = c/a (where c2 = a2 + b2)
  • Latus Rectum: 2b2/a
  • Directrix: y = ±a2/c
  • Asymptotes: y = ±(a/b)x

Example: Horizontal Hyperbola

Given the equation x2/9 – y2/25 = 1, determine:

  1. Equation of the asymptotes
  2. Coordinates of the foci
  3. Coordinates of the vertices
  4. Length of the latus rectum

Solution:

  1. Asymptotes: y = ±(5/3)x
  2. Foci: F (±√34, 0)
  3. Vertices: V (±3, 0)
  4. Latus Rectum: 50/3

Example Graph

Hyperbola with Center (h, k)

Horizontal Focal Axis

Horizontal Hyperbola with Center (h, k)

Equation: (x – h)2/a2 – (y – k)2/b2 = 1

Vertical Focal Axis

Vertical Hyperbola with Center (h, k)

Equation: (y – k)2/a2 – (x – h)2/b2 = 1

Example: Finding Center, Vertices, and Foci

Given the equation (x – 0)2/4 – (y – 1)2/9 = 1, find the center, coordinates of the vertices, and foci.

Solution:

  • Center: C (0, 1)
  • Vertices: V (-2, 1) and V (2, 1)
  • Foci: F (-√13, 1) and F (√13, 1)

Example: Finding the Equation of a Hyperbola

The center of a hyperbola is at point C (-2, 5) and its latus rectum is 4. If the length of its transverse axis is 8 and it is parallel to the x axis, determine its equation.

Solution:

Using the given information and the standard equation for a horizontal hyperbola, we can derive the equation: (x + 2)2/16 – (y – 5)2/4 = 1