All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

The ellipse

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

An ellipse is a stretched circle. In F3 you need its cartesian equation, its parametric equations and how to read off its key features.

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Cartesian equation

x2/a2 + y2/b2 = 1.

It crosses the x-axis at (±a, 0) and the y-axis at (0, ±b), and is symmetric in both axes with centre O.

If a > b the major axis (length 2a) lies along the x-axis and the minor axis (length 2b) along the y-axis. If b > a they swap. If a = b it is a circle of radius a.

Parametric equations

x = a cos t, y = b sin t, for 0 ≤ t < 2π.

Check: x2/a2 + y2/b2 = cos2 t + sin2 t = 1. The parameter t is not the angle the radius makes with the x-axis (unless a = b); it is the angle on the auxiliary circle of radius a.

Rearranging to standard form

Divide so the right-hand side is 1. For example 4x2 + 9y2 = 36 becomes x2/9 + y2/4 = 1, so a = 3, b = 2.

Worked example

Write 4x2 + 25y2 = 100 in standard form and give parametric equations.

  1. Divide by 100: x2/25 + y2/4 = 1
  2. a = 5 and b = 2

Answer: x2/25 + y2/4 = 1; x = 5 cos t, y = 2 sin t

Key idea: Ellipse: x2/a2 + y2/b2 = 1, with parametric form x = a cos t, y = b sin t. Intercepts (±a, 0) and (0, ±b).

Check yourself

  1. Find the length of the major axis of the ellipse 4x2 + 9y2 = 36.
    Show answer

    6. Standard form x2/9 + y2/4 = 1, so a = 3 and b = 2. The major axis has length 2a = 6.

  2. Which cartesian equation matches x = 5 cos t, y = 2 sin t?
    • x2/5 + y2/2 = 1
    • x2/4 + y2/25 = 1
    • x2/25 - y2/4 = 1
    • x2/25 + y2/4 = 1
    Show answer

    x2/25 + y2/4 = 1. cos t = x/5 and sin t = y/2, and cos2 t + sin2 t = 1.

  3. A point on the ellipse x = 4 cos t, y = 3 sin t has t = π/3. Find its y-coordinate, to 3 decimal places.
    Show answer

    2.598. y = 3 sin(π/3) = 3sqrt(3)/2 = 2.598.

  4. For x2/9 + y2/16 = 1, where is the major axis?
    • Along the x-axis, length 6
    • Along the y-axis, length 4
    • Along the y-axis, length 8
    • Along the x-axis, length 18
    Show answer

    Along the y-axis, length 8. b2 = 16 > a2 = 9, so the ellipse is longer in the y direction: from (0, -4) to (0, 4).

  5. The point (2, k), where k > 0, lies on x2/8 + y2/2 = 1. Find k.
    Show answer

    1. 4/8 + k2/2 = 1, so k2 = 1 and k = 1.

  6. When is the ellipse x2/a2 + y2/b2 = 1 a circle?
    • When a = 2b
    • When b = 0
    • When a = b
    • When a + b = 1
    Show answer

    When a = b. With a = b the equation becomes x2 + y2 = a2.

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Last updated 8 November 2026. Spotted a mistake? Tell us.