A LevelFurther MathsEdexcelFurther Pure 1Revision NotesCoordinate SystemsProperties of Ellipses, Parabolas & HyperbolasProperties of Parabolas

Properties of Parabolas (Edexcel A Level Further Maths): Revision Note

Exam code: 9FM0

Mark Curtis

Last updated

Properties of parabolas

What is a parabola?

  • A standard parabola is a curve with the Cartesian equation

    • y squared equals 4 a x

      • where a greater than 0

    • which looks like y equals x squared has been rotated 90° clockwise

      • Its line of symmetry is y equals 0 (the x-axis)

      • Its vertex is at open parentheses 0 comma space 0 close parentheses

Parabola graph with equation y^2 = 4ax, opening to the right (a C shape), intersecting the origin, with labelled x and y axes.

  • A parabola is one of the conic curves

    • with eccentricity e equals 1

Diagram of conic sections showing circles, ellipses, parabolas, and hyperbolas formed by intersecting a plane with cones.

Examiner Tips and Tricks

You are given the Cartesian equation of a parabola in the formulae booklet.

What are the parametric equations of a parabola?

  • The parametric equations of a parabola are

    • x equals a t squared

    • y equals 2 a t

    • where t element of straight real numbers

  • Eliminating the parameter, t, gives the Cartesian equation y squared equals 4 a x

Examiner Tips and Tricks

You are given the parametric equations of a parabola in the formulae booklet.

What are the coordinates of a general point on a parabola?

  • A general point P on the parabola y squared equals 4 a x has coordinates given by its parametric equations, P open parentheses a t squared comma space 2 a t close parentheses

Graph of parabola y^2 = 4ax (a C shape) with point P(at², 2at) marked on the curve, showing x and y axes, with parabola intersecting at origin O.

  • e.g. P open parentheses 3 t squared comma space 6 t close parentheses is a general point on the parabola y squared equals 12 x (where a equals 3)

    • It satisfies the equation of the curve

    • It moves around the curve depending on the value of t

  • This is different to, say, open parentheses 3 comma space 6 close parentheses

    • which is a fixed point on the parabola y squared equals 12 x

What is the eccentricity, focus and directrix of a parabola?

  • The eccentricity of a parabola, e, is always one

    • e equals 1

  • The focus, F, is the point open parentheses a comma space 0 close parentheses on the x-axis

  • The directrix is the vertical line with equation x equals negative a

The parabola (C shape) with equation y^2=4ax and the point F marked at (a, 0) and the vertical line x=-a drawn.

Examiner Tips and Tricks

You are given the eccentricity, focus and directrix of a parabola in the formulae booklet.

Worked Example

A parabola has the equation y squared equals 20 x.

Calculate

(a) the coordinates of the focus,

(b) the equation of the directrix.

Answer:

(a)

Find a by comparing to the general equation y squared equals 4 a x

a equals 5

Substitute into open parentheses a comma space 0 close parentheses

The focus has coordinates open parentheses 5 comma space 0 close parentheses

(b)

Substitute a equals 5 into the equation of a directrix, x equals negative a

The directrix has the equation x equals negative 5

What is the focus-directrix property of a parabola?

  • The focus-directrix property says that, if you take any point P on a parabola, then

    • the distance from P to the focus, F

    • divided by the shortest distance from P to the directrix (at point D)

    • is always equal to e, the eccentricity, where e equals 1

    • i.e. fraction numerator P F over denominator P D end fraction equals 1

      • sometimes rearranged to P F equals P D

A parabola (C shape) with the point F (a, 0) marked on the x-axis and the point P on the curve and the vertical line x=-a shown with the point D on the vertical line at the same height as P. The lines PF and PD are drawn. The formula PF/PD=1 is given.

Examiner Tips and Tricks

You are not given the focus-directrix property in the exam (you must learn it).

Worked Example

A parabola with focus F at open parentheses a comma space 0 close parenthesesand directrix x equals negative a is shown below.

The point P on the parabola has coordinates open parentheses x comma space y close parentheses and the point D is on the directrix, at the same height as P.

A parabola (C shape) with the point F (a, 0) marked on the x-axis and the point P(x,y) on the curve and the vertical line x=-a shown with the point D on the vertical line at the same height as P.

Using only the focus-directrix property, derive the Cartesian equation of a parabola, y squared equals 4 a x.

Answer:

Use the focus-directrix property on P, F and D (draw on the lines P F and P D)

fraction numerator P F over denominator P D end fraction equals 1

It helps to draw the lengths x and y from P open parentheses x comma space y close parentheses on the diagram

Create a right-angled triangle whose hypotenuse is P F with base open parentheses x minus a close parentheses and height y

A parabola (C shape) with the point F (a, 0) marked on the x-axis and the point P on the curve and the vertical line x=-a shown with the point D on the vertical line at the same height as P. The lines PF and PD are drawn. The formula PF/PD=1 is given. PF is the hypotenuse of a right-angled triangle with base (x-a) and height y. The length PD is shown as (x+a).

Use Pythagoras' theorem to find P F squared

P F squared equals open parentheses x minus a close parentheses squared plus y squared

Find the length P D from x to the directrix

P D equals x minus open parentheses negative a close parentheses equals x plus a

Rearrange fraction numerator P F over denominator P D end fraction equals 1 to make P F squared the subject

P F squared equals P D squared

Substitute in expressions for P F squared and P D squared from above

open parentheses x minus a close parentheses squared plus y squared equals open parentheses x plus a close parentheses squared

Expand and cancel

table row cell x squared minus 2 a x plus a squared plus y squared end cell equals cell x squared plus 2 a x plus a squared end cell row cell up diagonal strike x squared end strike minus 2 a x plus down diagonal strike a squared end strike plus y squared end cell equals cell up diagonal strike x squared end strike plus 2 a x plus down diagonal strike a squared end strike end cell row cell negative 2 a x plus y squared end cell equals cell 2 a x end cell end table

Add 2 a x to both sides

The Cartesian equation is y squared equals 4 a x

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Mark Curtis

Author: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.