Ellipse Worksheet

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  1. 1.What is the centre of (x − 2)²/25 + (y + 1)²/9 = 1?
  2. 2.What are the vertices (ends of the major axis) of (x − 2)²/25 + (y + 1)²/9 = 1?
  3. 3.Where are the foci of (x − 2)²/25 + (y + 1)²/9 = 1?
  4. 4.Is the major axis of x²/16 + y²/25 = 1 horizontal or vertical?
  5. 5.What are the vertices (ends of the major axis) of x²/16 + y²/25 = 1?
  6. 6.Where are the foci of x²/16 + y²/25 = 1?
  7. 7.How long are the major and minor axes of x²/25 + y²/9 = 1?
  8. 8.What are the vertices (ends of the major axis) of x²/25 + y²/9 = 1?
  9. 9.Where are the foci of x²/25 + y²/9 = 1?
  10. 10.What is the centre of (x + 3)²/16 + (y − 2)²/25 = 1?
  11. 11.What are the vertices (ends of the major axis) of (x + 3)²/16 + (y − 2)²/25 = 1?
  12. 12.Where are the foci of (x + 3)²/16 + (y − 2)²/25 = 1?
  13. 13.Is the major axis of (x − 1)²/169 + (y − 4)²/25 = 1 horizontal or vertical?
  14. 14.What are the vertices (ends of the major axis) of (x − 1)²/169 + (y − 4)²/25 = 1?
  15. 15.Where are the foci of (x − 1)²/169 + (y − 4)²/25 = 1?
  16. 16.How long are the major and minor axes of x²/36 + y²/100 = 1?
  17. 17.What are the vertices (ends of the major axis) of x²/36 + y²/100 = 1?
  18. 18.Where are the foci of x²/36 + y²/100 = 1?
  19. 19.What is the centre of (x + 2)²/100 + (y + 3)²/64 = 1?
  20. 20.What are the vertices (ends of the major axis) of (x + 2)²/100 + (y + 3)²/64 = 1?
  21. 21.Where are the foci of (x + 2)²/100 + (y + 3)²/64 = 1?
  22. 22.Is the major axis of (x − 4)²/9 + y²/25 = 1 horizontal or vertical?
  23. 23.What are the vertices (ends of the major axis) of (x − 4)²/9 + y²/25 = 1?
  24. 24.Where are the foci of (x − 4)²/9 + y²/25 = 1?
  25. 25.How long are the major and minor axes of x²/169 + (y − 5)²/144 = 1?
  26. 26.What are the vertices (ends of the major axis) of x²/169 + (y − 5)²/144 = 1?
  27. 27.Where are the foci of x²/169 + (y − 5)²/144 = 1?
  28. 28.What is the centre of (x + 1)²/64 + (y − 1)²/289 = 1?
  29. 29.What are the vertices (ends of the major axis) of (x + 1)²/64 + (y − 1)²/289 = 1?
  30. 30.Where are the foci of (x + 1)²/64 + (y − 1)²/289 = 1?
  31. 31.Is the major axis of (x − 3)²/25 + (y − 3)²/16 = 1 horizontal or vertical?
  32. 32.What are the vertices (ends of the major axis) of (x − 3)²/25 + (y − 3)²/16 = 1?
  33. 33.Where are the foci of (x − 3)²/25 + (y − 3)²/16 = 1?
  34. 34.How long are the major and minor axes of x²/100 + (y + 2)²/36 = 1?
  35. 35.What are the vertices (ends of the major axis) of x²/100 + (y + 2)²/36 = 1?
  36. 36.Where are the foci of x²/100 + (y + 2)²/36 = 1?
  37. 37.An ellipse has centre (0, 0), a vertex at (5, 0) and a focus 4 units from the centre on the same axis. What is its equation?
  38. 38.An ellipse has centre (0, 0), a vertex at (0, 5) and a focus 3 units from the centre on the same axis. What is its equation?
  39. 39.An ellipse has centre (1, −2), a vertex at (14, −2) and a focus 12 units from the centre on the same axis. What is its equation?
  40. 40.An ellipse has centre (−4, 3), a vertex at (−4, 13) and a focus 8 units from the centre on the same axis. What is its equation?
  41. 41.An ellipse has centre (2, 2), a vertex at (2, 7) and a focus 4 units from the centre on the same axis. What is its equation?
  42. 42.An ellipse has centre (0, −1), a vertex at (17, −1) and a focus 15 units from the centre on the same axis. What is its equation?
  43. 43.For any point on an ellipse, the sum of its distances to the two foci equals…
  44. 44.In an ellipse, how are a, b and c related?
  45. 45.What is the eccentricity of an ellipse?
  46. 46.An ellipse has a = 5 and b = 5. What shape is it?
  47. 47.What is the area of the ellipse x²/25 + y²/9 = 1?
  48. 48.Which ellipse is the most stretched out: e = 0.2, 0.5, 0.8 or 0.95?
How long are the major and minor axes of x²/25 + y²/9 = 1?

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Unofficial practice questions, not affiliated with any school, textbook publisher or worksheet maker. Bank checked September 23, 2026.

Next step More precalculus: trig identitiesNext: the unit circle, the ellipse with a = b = 1

How it works

This ellipse worksheet has 48 problems on ellipses in standard form: find the centre, the direction and length of the axes, the vertices and the foci, then write the equation from given points.

Read the full guide

Four sets

Centre and axes reads h, k, a and b from the equation. Vertices and foci works out the key points, using c² = a² − b². Write the equation goes the other way. The ideas set covers eccentricity, area and the definition.

Practice or print

Practice brings a missed problem back until you get it right. The sheet on screen is what prints: remove a question, add more up to all 48, and tick With answer key.

  • The larger denominator is a²; the variable it sits under tells you the direction of the major axis.
  • Vertices are a from the centre, foci are c from the centre, and both lie on the major axis.

Questions and answers

How do you find the foci of an ellipse?

Take a² as the larger denominator and b² as the smaller, then c² = a² − b². The foci are c units from the centre along the major axis, one on each side.

How do you tell if an ellipse is horizontal or vertical?

Compare the denominators. If the larger one is under the x term, the major axis is horizontal; if it is under the y term, the major axis is vertical.

What is the eccentricity of an ellipse?

e = c/a, the distance from centre to focus divided by the distance from centre to vertex. It is 0 for a circle and approaches 1 as the ellipse flattens.

All 48 questions with answers

Centre and axes 12

  • How long are the major and minor axes of x²/25 + y²/9 = 1? Answer: 10 and 6. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • How long are the major and minor axes of x²/36 + y²/100 = 1? Answer: 20 and 12. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • How long are the major and minor axes of x²/100 + (y + 2)²/36 = 1? Answer: 20 and 12. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • How long are the major and minor axes of x²/169 + (y − 5)²/144 = 1? Answer: 26 and 24. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Is the major axis of (x − 1)²/169 + (y − 4)²/25 = 1 horizontal or vertical? Answer: Horizontal. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Is the major axis of (x − 3)²/25 + (y − 3)²/16 = 1 horizontal or vertical? Answer: Horizontal. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Is the major axis of (x − 4)²/9 + y²/25 = 1 horizontal or vertical? Answer: Vertical. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Is the major axis of x²/16 + y²/25 = 1 horizontal or vertical? Answer: Vertical. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What is the centre of (x + 1)²/64 + (y − 1)²/289 = 1? Answer: (−1, 1). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What is the centre of (x + 2)²/100 + (y + 3)²/64 = 1? Answer: (−2, −3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What is the centre of (x + 3)²/16 + (y − 2)²/25 = 1? Answer: (−3, 2). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What is the centre of (x − 2)²/25 + (y + 1)²/9 = 1? Answer: (2, −1). Source: Wikipedia, Ellipse. Read 2026-09-23.
Show all 48 questions (36 more)

Vertices and foci 24

  • What are the vertices (ends of the major axis) of (x + 1)²/64 + (y − 1)²/289 = 1? Answer: (−1, 18) and (−1, −16). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of (x + 2)²/100 + (y + 3)²/64 = 1? Answer: (8, −3) and (−12, −3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of (x + 3)²/16 + (y − 2)²/25 = 1? Answer: (−3, 7) and (−3, −3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of (x − 1)²/169 + (y − 4)²/25 = 1? Answer: (14, 4) and (−12, 4). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of (x − 2)²/25 + (y + 1)²/9 = 1? Answer: (7, −1) and (−3, −1). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of (x − 3)²/25 + (y − 3)²/16 = 1? Answer: (8, 3) and (−2, 3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of (x − 4)²/9 + y²/25 = 1? Answer: (4, 5) and (4, −5). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of x²/16 + y²/25 = 1? Answer: (0, 5) and (0, −5). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of x²/25 + y²/9 = 1? Answer: (5, 0) and (−5, 0). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of x²/36 + y²/100 = 1? Answer: (0, 10) and (0, −10). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of x²/100 + (y + 2)²/36 = 1? Answer: (10, −2) and (−10, −2). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What are the vertices (ends of the major axis) of x²/169 + (y − 5)²/144 = 1? Answer: (13, 5) and (−13, 5). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x + 1)²/64 + (y − 1)²/289 = 1? Answer: (−1, 16) and (−1, −14). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x + 2)²/100 + (y + 3)²/64 = 1? Answer: (4, −3) and (−8, −3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x + 3)²/16 + (y − 2)²/25 = 1? Answer: (−3, 5) and (−3, −1). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x − 1)²/169 + (y − 4)²/25 = 1? Answer: (13, 4) and (−11, 4). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x − 2)²/25 + (y + 1)²/9 = 1? Answer: (6, −1) and (−2, −1). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x − 3)²/25 + (y − 3)²/16 = 1? Answer: (6, 3) and (0, 3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of (x − 4)²/9 + y²/25 = 1? Answer: (4, 4) and (4, −4). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of x²/16 + y²/25 = 1? Answer: (0, 3) and (0, −3). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of x²/25 + y²/9 = 1? Answer: (4, 0) and (−4, 0). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of x²/36 + y²/100 = 1? Answer: (0, 8) and (0, −8). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of x²/100 + (y + 2)²/36 = 1? Answer: (8, −2) and (−8, −2). Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Where are the foci of x²/169 + (y − 5)²/144 = 1? Answer: (5, 5) and (−5, 5). Source: Wikipedia, Ellipse. Read 2026-09-23.

Write the equation 6

  • An ellipse has centre (0, 0), a vertex at (0, 5) and a focus 3 units from the centre on the same axis. What is its equation? Answer: x²/16 + y²/25 = 1. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • An ellipse has centre (0, 0), a vertex at (5, 0) and a focus 4 units from the centre on the same axis. What is its equation? Answer: x²/25 + y²/9 = 1. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • An ellipse has centre (0, −1), a vertex at (17, −1) and a focus 15 units from the centre on the same axis. What is its equation? Answer: x²/289 + (y + 1)²/64 = 1. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • An ellipse has centre (1, −2), a vertex at (14, −2) and a focus 12 units from the centre on the same axis. What is its equation? Answer: (x − 1)²/169 + (y + 2)²/25 = 1. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • An ellipse has centre (2, 2), a vertex at (2, 7) and a focus 4 units from the centre on the same axis. What is its equation? Answer: (x − 2)²/9 + (y − 2)²/25 = 1. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • An ellipse has centre (−4, 3), a vertex at (−4, 13) and a focus 8 units from the centre on the same axis. What is its equation? Answer: (x + 4)²/36 + (y − 3)²/100 = 1. Source: Wikipedia, Ellipse. Read 2026-09-23.

The ideas 6

  • An ellipse has a = 5 and b = 5. What shape is it? Answer: A circle. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • For any point on an ellipse, the sum of its distances to the two foci equals… Answer: 2a, the length of the major axis. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • In an ellipse, how are a, b and c related? Answer: c² = a² − b². Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What is the area of the ellipse x²/25 + y²/9 = 1? Answer: 15π. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • What is the eccentricity of an ellipse? Answer: c/a. Source: Wikipedia, Ellipse. Read 2026-09-23.
  • Which ellipse is the most stretched out: e = 0.2, 0.5, 0.8 or 0.95? Answer: e = 0.95. Source: Wikipedia, Ellipse. Read 2026-09-23.

How this list was compiled: Questions, wrong choices and explanations were written for this site. The standard form, c² = a² − b², the vertices and foci on the major axis, the sum-of-distances definition, eccentricity c/a and the area πab were checked on Wikipedia's Ellipse article (https://en.wikipedia.org/wiki/Ellipse), read 2026-09-23. Every centre, vertex, focus and equation was worked out from whole-number a, b and c. Licence: own work

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