Hyperbola Math Example 2

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Example 2

medium
Find the equations of the asymptotes for the hyperbola y24−x29=1\frac{y^2}{4} - \frac{x^2}{9} = 1.

Solution

  1. 1
    This is a vertical hyperbola with a2=4a^2 = 4 (under y2y^2) and b2=9b^2 = 9, so a=2a = 2 and b=3b = 3.
  2. 2
    For a vertical hyperbola y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1, the asymptotes are y=±abxy = \pm \frac{a}{b} x.
  3. 3
    The asymptotes are y=±23xy = \pm \frac{2}{3} x.

Answer

y=23xandy=−23xy = \frac{2}{3}x \quad \text{and} \quad y = -\frac{2}{3}x
The asymptotes of a hyperbola are the lines that the curve approaches but never touches as it extends to infinity. For a vertical hyperbola, the slopes are ±a/b\pm a/b; for a horizontal one, ±b/a\pm b/a. The asymptotes pass through the center and guide the shape of the curve.

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 3 medium

Find the foci of the hyperbola [formula].

Example 4 hard

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

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