The Baskara Formula as a Property of Conic

A practical way to obtain a parabola is by obliquely sectioning a straight circular cone

Such a curve thus obtained has a certain property that characterizes it. This property says that every point belonging to it will be equidistant from a fixed line and a fixed point. This can be written algebraically through a number of forms, but highlight two:

The reduced form

The Polynomial Form

With the help of a graph, it is possible to observe other properties that condition such expressions

P: is a fixed point of the plane that contains the parabola.

X: is an auxiliary axis that symmetrically divides the parabola, it passes through the point P (the focal point of the parabola) .

V: it is a point belonging to the parabola and to the auxiliary axis, it is the vertex.

f: It is a fixed auxiliary line, it is the directrix of the parabola.

B: is a generic point belonging to the parabola, the parabola is a geometrical place that satisfies the following condition regarding the measurements of the segments:

p: corresponds to the PV segment measurement:

Comparing both forms, we extract the following relationships:

Now let’s redraw the parabola in the Cartesian plane by positioning point B on the axis of the abscissas and introducing new visual elements

Using Girard’s relations we can obtain the abscissa of the point P′

and consequently the ordinate of point V

and the measure of PV′ will be

With this we have enough elements to apply the Pythagorean theorem to the triangle BPP′

and solving for “x” we get the expression of the well-known Baskara formula: