An alternative way to define a parabola is as a locus of points.
Focus and directrix
The locus defining a parabola depends on a focus and a directrix.
The focus is a point. For a standard parabola, the focus is located on the x axis a distance $a$ from the origin, that is at the point (a, 0). $a$ is the constant in the parabola equation $y^2 = 4 a x$
The directrix is a line. For a standard parabola it is a line perpendicular to the x axis passing through (-a, 0), that is the line $x = -a$
The vertex of the parabola is the turning point. It is always halfway between the focus and directrix, which is always at the origin for a standard parabola.
This diagram shows the focus, directrix and vertex for the case $a = 1$:
A parabola is the locus of all points which are an equal distance from the focus and directrix:
$$ FP = PD $$
The animation below illustrates this:
Join the GraphicMaths Newletter
Sign up using this form to receive an email when new content is added:
angle cartesian equation chord circle combinations cosh cosine cosine rule cube diagonal directrix ellipse equilateral triangle exponential exterior angle focus horizontal hyperbola hyperbolic function interior angle inverse hyperbolic function isosceles triangle locus major axis minor axis normal parabola parametric equation permutations power quadrilateral radius root sine rule sinh sloping lines solving equations solving triangles square standard curves straight line graphs tangent tanh transformations triangle vertical