A circle is a locus of points whose distance from a fixed point is a constant. A parabola can also be described as a locus of points whose distance from a fixed point and a fixed line not passing through that point is a constant. An example of a parabola is shown below.
In the figure below, point is called the focus of the parabola and line is called its directrix. The vertex of the parabola is at the origin O and the -axis the perpendicular bisector of . If we take any point on the parabola, draw , and draw perpendicular to line where Q is line l, then the distance between and and and are equal.
Suppose that the coordinates of the focus is where , then the directrix is (can you see why?).
From here we can see that and . Since is equal to ,
Squaring both sides, we have
This is the equation of the parabola with focus and directrix