Algebra 2 · Unit 8 · Topic 8.4
Ellipses and Their Axes
An axis-aligned ellipse has equation (x − h)²/a² + (y − k)²/b² = 1 with a, b > 0.
Definition and explanation
An axis-aligned ellipse has equation (x − h)²/a² + (y − k)²/b² = 1 with a, b > 0. The denominators are squared semiaxis lengths; the larger denominator indicates the major-axis direction.
Worked example
For x²/25 + y²/9 = 1, the center is (0, 0), horizontal semiaxis is 5, and vertical semiaxis is 3. The vertices on the major axis are (±5, 0). The foci are (±4, 0) because 25 − 9 = 16.
Common mistake
The full major-axis length is 10, not the semiaxis length 5.
Key takeaway
An axis-aligned ellipse has equation (x − h)²/a² + (y − k)²/b² = 1 with a, b > 0.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.