Hyperbola Math Example 4

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Example 4

hard
Write the equation of the hyperbola with foci at (0,±5)(0, \pm 5) and vertices at (0,±3)(0, \pm 3).

Solution

  1. 1
    Foci on the yy-axis means vertical hyperbola: y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1. Vertices give a=3a = 3, foci give c=5c = 5.
  2. 2
    b2=c2−a2=25−9=16b^2 = c^2 - a^2 = 25 - 9 = 16. The equation is y29−x216=1\frac{y^2}{9} - \frac{x^2}{16} = 1.

Answer

y29−x216=1\frac{y^2}{9} - \frac{x^2}{16} = 1
From vertices and foci, we get aa and cc directly. Since c2=a2+b2c^2 = a^2 + b^2 for hyperbolas, we find b2=c2−a2b^2 = c^2 - a^2. The position of the foci determines whether the hyperbola is horizontal or vertical.

About Hyperbola

The set of all points in a plane where the absolute difference of the distances to two fixed points (foci) is constant. The curve has two separate branches and asymptotes.

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More Hyperbola Examples

Example 1 easy

Identify the vertices and the direction of opening for the hyperbola [formula].

Example 2 medium

Find the equations of the asymptotes for the hyperbola [formula].

Example 3 medium

Find the foci of the hyperbola [formula].

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