Hyperbola Math Example 3

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Example 3

medium
Find the foci of the hyperbola x225−y2144=1\frac{x^2}{25} - \frac{y^2}{144} = 1.

Solution

  1. 1
    For a hyperbola, c2=a2+b2c^2 = a^2 + b^2 (note: addition, unlike ellipses). Here c2=25+144=169c^2 = 25 + 144 = 169, so c=13c = 13.
  2. 2
    Since the transverse axis is horizontal, the foci are at (±13,0)(\pm 13, 0).

Answer

Foci at (±13,0)\text{Foci at } (\pm 13, 0)
Unlike ellipses where c2=a2−b2c^2 = a^2 - b^2, hyperbolas use c2=a2+b2c^2 = a^2 + b^2. This is because the defining property of a hyperbola is the difference (not sum) of distances to the foci. The foci always lie on the transverse axis, beyond the vertices.

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 4 hard

Write the equation of the hyperbola with foci at [formula] and vertices at [formula].

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