Derivation of the Equation of the Parabola

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 F is called the focus of the parabola and line l is called its directrix. The vertex of the parabola is at the origin O and the x-axis the perpendicular bisector of FH. If we take any point P(x,y) on the parabola, draw FP, and draw PQ perpendicular to line l where Q is line l, then the distance between F and P and P and Q are equal.



Suppose that the coordinates of the focus is (0,p) where p > 0, then the directrix is y = -p (can you see why?).

From here we can see that PQ = |y + p| and PF = \sqrt{x^2 + (y - p)^2 }. Since PQ is equal to PF,

|y + p| = \sqrt{x^2 + (y - p)^2 }

Squaring both sides, we have

(y + p)^2 = x^2 + (y-p)^2
y^2 + 2py + p^2 = x^2 + y^2 - 2py + y^2
2py = x^2 - 2py
4py = x^2
y = \frac{1}{4p}x^2.

This is the equation of the parabola with focus (0,p) and directrix y = -p.

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