Suppose the point P has Cartesian coordinates and l is a horizontal line with the equation . We can do this without loss of generality by simply rotating the coordinate system appropriately. Now choose any point in the plane with the general coordinates .
The distance from P to is given by .
The distance from l to is given by .
So, equating these distances and solving for y, we find:
Since all of the indexed variables are constants, this is simply a quadratic equation, proving that the locus of points equidistant from P and l forms a parabola.