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The parabola y2 = 4ax

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

In F1 the parabola is written with x and y swapped from the familiar y = x2: y2 = 4ax opens to the right. You need its parametric form and its focus-directrix property.

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Cartesian and parametric forms

y2 = 4ax (a > 0) is symmetric about the x-axis with vertex at O.

Its parametric form is x = at2, y = 2at. Check: y2 = 4a2t2 = 4a(at2) = 4ax. So (at2, 2at) is a general point on the curve, with one value of t for each point.

Focus and directrix

The point S(a, 0) is the focus and the line x = -a is the directrix. The parabola is the locus of points equidistant from the focus and the directrix.

Proof: for P(x, y), PS2 = (x - a)2 + y2 and the distance to the directrix is x + a. Setting (x - a)2 + y2 = (x + a)2 and expanding gives y2 = 4ax.

Using the property

The distance from a point P on the parabola to the focus equals its distance to the directrix: PS = x + a. This turns distance questions into one subtraction. Give a focus as coordinates, (a, 0), and a directrix as an equation, x = -a.

Worked example

A point P moves so that it is equidistant from (4, 0) and the line x = -4. Find the equation of its locus.

  1. PS2 = (x - 4)2 + y2 and the distance to the line is x + 4.
  2. (x - 4)2 + y2 = (x + 4)2
  3. x2 - 8x + 16 + y2 = x2 + 8x + 16, so y2 = 16x.

Answer: y2 = 16x, a parabola with a = 4

Key idea: y2 = 4ax: parametric (at2, 2at), focus (a, 0), directrix x = -a. Every point is as far from the focus as from the directrix.

Check yourself

  1. The parabola y2 = 12x has focus (a, 0). Find a.
    Show answer

    3. 4a = 12, so a = 3 and the focus is (3, 0).

  2. What is the directrix of y2 = 20x?
    • x = 5
    • y = -5
    • x = -20
    • x = -5
    Show answer

    x = -5. 4a = 20 so a = 5, and the directrix is x = -a = -5.

  3. The point with parameter t = 3 lies on the parabola y2 = 8x. Find its y-coordinate.
    Show answer

    12. a = 2, so the point is (at2, 2at) = (18, 12).

  4. P lies on y2 = 8x and has y-coordinate 8. Find the distance from P to the focus.
    Show answer

    10. At y = 8: 64 = 8x, so x = 8. a = 2, so PS = x + a = 10.

  5. A curve has parametric equations x = 5t2, y = 10t. Find the x-coordinate of its focus.
    Show answer

    5. y2 = 100t2 = 20(5t2) = 20x, so 4a = 20, a = 5, focus (5, 0).

  6. A point is equidistant from (3, 0) and the line x = -3. What is its locus?
    • y2 = 3x
    • x2 = 12y
    • y2 = 12x
    • y2 = 6x
    Show answer

    y2 = 12x. Focus (a, 0) and directrix x = -a with a = 3 give y2 = 4ax = 12x.

  7. Which point lies on y2 = 4ax for every value of t?
    • (at2, 2at)
    • (2at, at2)
    • (at, 2at2)
    • (a t2, at)
    Show answer

    (at2, 2at). (2at)2 = 4a2t2 = 4a(at2).

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