The parabola is thus identified with the curve of the same name studied in treatises on analytic geometry.
120. Equation of central conics referred to conjugate diameters. Consider now a central conic, that is, one which is not a parabola and the center of which is therefore at a finite distance. Draw any four tangents to it, two of which are parallel (Fig. 31). Let the parallel tangents meet one of the other tangents in A and B and the other in C and D, and let P and Q be the points of contact of the parallel tangents R and S of the others. Then AC, BD, PQ, and RS all meet in a point W (§ 88). From the figure,
PW : WQ = AP : QC = PD : BQ,
or
AP · BQ = PD · QC.
If now DC is a fixed tangent and AB a variable one, we have from this equation
AP · BQ = constant.
This constant will be positive or negative according as PA and BQ are measured in the same or in opposite directions. Accordingly we write
AP · BQ = ± b2.