Ellipse Examples in Math

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

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 sum of the distances to two fixed points (foci) is constant. Standard form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

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: An ellipse generalizes a circle by having two axes of different lengths. The relationship c^2 = a^2 - b^2 connects the foci to the axes. When a = b, the ellipse becomes a circle.

Common stuck point: The larger denominator is always a^2, regardless of whether it's under x or y. If a^2 is under y, the major axis is vertical.

Sense of Study hint: Compare the two denominators. The larger one is a^2 and tells you which direction the major axis goes. Then find c using c^2 = a^2 - b^2.

Worked Examples

Example 1

easy
Find the lengths of the semi-major and semi-minor axes of the ellipse \frac{x^2}{25} + \frac{y^2}{9} = 1.

Solution

  1. 1
    The standard form is \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 where a > b > 0.
  2. 2
    Here a^2 = 25 and b^2 = 9, so a = 5 and b = 3.
  3. 3
    The semi-major axis has length a = 5 (along the x-axis) and the semi-minor axis has length b = 3 (along the y-axis).

Answer

a = 5 \text{ (semi-major)}, \quad b = 3 \text{ (semi-minor)}
An ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 has semi-major axis a (the larger denominator's square root) and semi-minor axis b. The major axis lies along whichever variable has the larger denominator.

Example 2

medium
Find the foci of the ellipse \frac{x^2}{16} + \frac{y^2}{25} = 1.

Practice Problems

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

Example 1

medium
Write the equation of an ellipse centered at the origin with foci at (\pm 4, 0) and a major axis of length 10.

Example 2

hard
Find the eccentricity of the ellipse \frac{(x-2)^2}{36} + \frac{(y+1)^2}{20} = 1 and describe what it tells us about the shape.
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Background Knowledge

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

equation of circle