Hyperbola Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Identify the vertices and the direction of opening for the hyperbola x29−y216=1\frac{x^2}{9} - \frac{y^2}{16} = 1.

Solution

  1. 1
    The standard form x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 opens left and right (horizontally).
  2. 2
    a2=9a^2 = 9, so a=3a = 3. The vertices are at (±a,0)=(±3,0)(\pm a, 0) = (\pm 3, 0).
  3. 3
    The transverse axis is along the xx-axis with vertices at (−3,0)(-3, 0) and (3,0)(3, 0).

Answer

Vertices: (±3,0); opens left and right\text{Vertices: } (\pm 3, 0); \text{ opens left and right}
In the standard form of a hyperbola, the positive fraction determines the direction of opening. When x2x^2 is positive, the hyperbola opens horizontally; when y2y^2 is positive, it opens vertically. The vertices are at distance aa from the center along the transverse axis.

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.

Learn more about Hyperbola →

More Hyperbola Examples

Example 2 medium

Find the equations of the asymptotes 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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