Edexcel IAL - Further Pure Mathematics 1- 4.3 Focus–Directrix Property of the Parabola- Study notes  - New syllabus

Focus–Directrix Property of the Parabola

A parabola is not just a curve given by an equation. It is defined geometrically as a locus of points.

Definition:

A parabola is the set of all points in a plane which are equidistant from a fixed point and a fixed straight line.

The fixed point is called the focus and the fixed straight line is called the directrix.

Standard Parabola

Consider the parabola:

\( y^2 = 4ax \)

For this parabola:

• The focus is at \( (a,0) \).
• The directrix is the vertical line \( x = -a \).
• The axis of symmetry is the x-axis.
• The vertex is at the origin \( (0,0) \).

Focus–Directrix Condition

Let \( P(x,y) \) be any point on the parabola.

Distance of \( P \) from the focus \( (a,0) \) is:

\( PF = \sqrt{(x-a)^2 + y^2} \)

Distance of \( P \) from the directrix \( x = -a \) is the perpendicular distance:

\( PD = |x + a| \)

For points on the parabola, these distances are equal:

\( PF = PD \)

Squaring both sides gives:

\( (x-a)^2 + y^2 = (x + a)^2 \)

Expanding:

\( x^2 – 2ax + a^2 + y^2 = x^2 + 2ax + a^2 \)

Canceling terms:

\( y^2 = 4ax \)

This shows that the equation of the parabola follows directly from the focus–directrix definition.

Geometric Meaning

Every point on the parabola is exactly as far from the focus as it is from the directrix. This gives the parabola its reflective property, which is used in satellite dishes, car headlights and telescopes.