60
ting the axis of ordinates in K, we have
_AK : OQ' = AC' : CC' = y : y',_
and
_OM : AK = BB' : AB' = y : y',_
and, by multiplication,
_OM : OQ' = y__2__ : y'__2__,_
or
_x : x' = y__2__ : y'__2__;_
whence
_The abscissas of two points on a parabola are to each other as the squares of the corresponding coördinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference._
The last equation may be written
_y__2__ = 2px,_
where _2p_ stands for _y'__2__ : x'_.
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 = ± b__2__._
[Figure 31]
FIG. 31
Since AD and BC are parallel tangents, PQ is a diameter and the conjugate diameter is parallel to AD. The middle point of PQ is the center of the coni