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 conic.  We take now for the axis of abscissas the diameter PQ, and the conjugate diameter for the axis of ordinates.  Join A to Q and B to P and draw a line through S parallel to the axis of ordinates.  These three lines all meet in a point N, because AP, BQ, and AB form a triangle circumscribed to the conic.  Let NS meet PQ in M.  Then, from the properties of the circumscribed triangle (§ 89), M, N, S, and the point at infinity on NS are four harmonic points, and therefore N is the middle point of MS.  If the cooerdinates of S are (x, y), so that OM is x and MS is y, then MN = y/2.  Now from the similar triangles PMN and PQB we have

BQ :  PQ = NM :  PM,

and from the similar triangles PQA and MQN,

AP :  PQ = MN :  MQ,

whence, multiplying, we have

_+-b__2__/4 a__2__ = y__2__/4 (a + x)(a — x),_

where

[formula]

or, simplifying,

[formula]

which is the equation of an ellipse when b_2_ has a positive sign, and of a hyperbola when b_2_ has a negative sign.  We have thus identified point-rows of the second order with the curves given by equations of the second degree.