Writing the Equation of a Parabola
The equation of a can be determined if its and is known.
Given any point on the parabola, the distance from that point to the focus is equal to the closest point on the directrix. Then, by using the , the lengths can be compared to get the equation for the parabola.
Any point on a parabola has the same distance to the , as to the closest point, on the . Thus, line segments are drawn from an point, on the parabola, to and
The length of each segment can be expressed with the since it's the distance between two points.
The coordinates for the focus can be read from the coordinate plane. Since the arbitrary point is any point on the parabola, its coordinates are unknown. Thus, the point
on the directrix, shares the same
and has the
The points can now be substituted into the expressions for the lengths.
The same thing is now done for the length
The lengths are now expressed as:
are equidistant to
has the same length. Thus, they can be put equal to each other.
The equation for the parabola can now be found by solving the equation for
Note that it is possible to expand
but it's easier to leave it as the equation is solved for
Therefore, the equation for the parabola is