The equation of the hyperbola, referred to the asymptotes as axes, is _xy =__ constant._

This identifies the curve with the hyperbola as defined and discussed in works on analytic geometry.

[Figure 30]

FIG. 30

119.  Equation of parabola. We have defined the parabola as a conic which is tangent to the line at infinity (§ 110).  Draw now two tangents to the curve (Fig. 30), meeting in A, the points of contact being B and C.  These two tangents, together with the line at infinity, form a triangle circumscribed about the conic.  Draw through B a parallel to AC, and through C a parallel to AB.  If these meet in D, then AD is a diameter.  Let AD meet the curve in P, and the chord BC in Q. P is then the middle point of AQ.  Also, Q is the middle point of the chord BC, and therefore the diameter AD bisects all chords parallel to BC.  In particular, AD passes through P, the point of contact of the tangent drawn parallel to BC.

Draw now another tangent, meeting AB in B’ and AC in C’.  Then these three, with the line at infinity, make a circumscribed quadrilateral.  But, by Brianchon’s theorem applied to a quadrilateral (§ 88), it appears that a parallel to AC through B’, a parallel to AB through C’, and the line BC meet in a point D’.  Also, from the similar triangles BB’D’ and BAC we have, for all positions of the tangent line B’C,

B’D’ :  BB’ = AC :  AB,

or, since B’D’ = AC’,

AC’:  BB’ = AC:AB = constant.

If another tangent meet AB in B" and AC in C", we have

_ AC’ :  BB’ = AC” :  BB”, _

and by subtraction we get

C’C” :  B’B” = constant;

whence

The segments cut off on any two tangents to a parabola by a variable tangent are proportional.

If now we take the tangent B’C’ as axis of ordinates, and the diameter through the point of contact O as axis of abscissas, calling the coordinates of B(x, y) and of C(x’, y’), then, from the similar triangles BMD’ and we have

y :  y’ = BD’ :  D’C = BB’ :  AB’.

Also

y :  y’ = B’D’ :  C’C = AC’ :  C’C.

If now a line is drawn through A parallel to a diameter, meeting 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 cooerdinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference.