Parabola Equation
by Emily Kennedy

Let's describe our construction algebraically.

Let (a,b) be the coordinates of the focus F, and let y = y₀ be the equation describing the directrix d (for simplicity, let's only look at horizontal directrices).

Finally, let (x₀,y₀) be the coordinates of X.
(Its y-coordinate must be y₀, since it lies on d.)

1: Draw

We will only need the slope m of this line. We have:

if a ≠ x₀.
(If a = x₀, then m is undefined because is vertical.)

2. Let M be the midpoint of .
Draw h, the perpendicular bisector of .

h goes through M, whose coordinates are , by the midpoint formula.
Also, h must have slope
(Note that this slope is 0 if a = x₀, which is just what we want--
a line perpendicular to a vertical line has slope 0.)
Thus, h is the line described by

3. Draw k, the line which is perpendicular to d and goes through X.

d is horizontal, so k must be vertical.
And k goes through the point X = (x₀,y₀).
Thus, k is the line described by
x = x₀

4. Let P be the point at which h and k intersect.
(This point exists because F is not on d.)

The intersection of h and k is simply the point on h when x = x₀.

This point is (x₀,y_i), where:

Now, letting the point X vary on d (i.e., letting x₀ vary in R), we have that the parabola is the set of points:

So the parabola is the curve in the plane described by

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