Hyperbola Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Hyperbola.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

While an ellipse keeps the SUM of distances to foci constant, a hyperbola keeps the DIFFERENCE constant. This creates two separate curves that open away from each other, each curving toward (but never reaching) a pair of asymptotic lines.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A hyperbola has two branches that approach asymptotes. The key relationship is c^2 = a^2 + b^2 (contrast with the ellipse's c^2 = a^2 - b^2). The positive term determines which direction it opens.

Common stuck point: Which way does it open? The POSITIVE variable's term tells you: positive x^2 term means horizontal opening, positive y^2 term means vertical opening.

Sense of Study hint: Draw the central rectangle using a and b, then draw the asymptotes as diagonals of that rectangle. The hyperbola hugs those asymptotes.

Worked Examples

Example 1

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

Solution

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

Answer

\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 x^2 is positive, the hyperbola opens horizontally; when y^2 is positive, it opens vertically. The vertices are at distance a from the center along the transverse axis.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

Example 2

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

These ideas may be useful before you work through the harder examples.

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