Diameter of a Hyperbola

I. Introduction to Diameter:

  • In geometry, the diameter of a geometric figure is a line segment connecting two points on the curve and passing through the center. For a hyperbola, the concept of a diameter is specific and noteworthy.

II. Diameter of a Hyperbola:

  1. Definition:

    • The diameter of a hyperbola is a line segment passing through the center and having both endpoints on the hyperbola.
    • It is the longest chord of the hyperbola.
  2. Length:

    • The length of the diameter is twice the length of the semi-major axis.
    • For a hyperbola with the equation x2a2y2b2=1, the length of the diameter is 2a.

III. Properties and Characteristics:

  1. Symmetry:

    • The diameter is the axis of symmetry for a hyperbola. It divides the hyperbola into two symmetrical branches.
  2. Foci Relationship:

    • The foci of the hyperbola are located on the major axis, and the diameter passes through both foci.
    • The sum of the distances from any point on the hyperbola to the foci is constant and equal to the length of the major axis.

Example: 

Consider the hyperbola given by the equation:

x216y29=1

Let's find the diameter of this hyperbola.

1. Identify Key Parameters:

  • The given hyperbola is in the form x2a2y2b2=1.
  • In this case, a=4 (semi-major axis) and b=3 (semi-minor axis).

2. Calculate the Diameter:

  • The length of the diameter (D) is twice the length of the semi-major axis (a): D=2a

  • Substitute a=4 into the formula: D=2×4=8

3. Interpretation:

  • The diameter of the given hyperbola is 8.

4. Verification:

  • We can verify this by choosing two points on the hyperbola that lie on a line passing through the center and calculating the distance between them.

  • For instance, consider two points: A(4,3) and B(4,3). These points are symmetric with respect to the center at the origin.

  • The distance between these points using the distance formula: AB=(44)2+(33)2=64+36=100=10 As expected, AB is equal to the length of the diameter (D).