There are two equivalent ways of finding the distance from a point to the parabola .
Firstly, we could us the distance formula from A to a point on the parabola:
Then
So, noting that the denominator is never zero unless the point is on the parabola, we only have to look at where the third degree equation on the numerator is equal to zero.
Alternatively, we can assume that the segment to the shortest-distance point on the parabola will have perpendicular slope to its tangent there. Since slopes on the parabola are given by , the lines will be given by:
Which is equivalent to the numerator of the expression above having value 0. The resulting zeros, and the distances, are graphed below.