An Elementary Course in Synthetic Projective Geometry Part 11

Therefore

_The asymptotes cut off on each tangent a segment which is bisected by the point of contact._

*114.* If we draw a line _OQ_ parallel to _AB_, then _OP_ and _OQ_ are conjugate diameters, since _OQ_ is parallel to the tangent at the point where _OP_ meets the curve. Then, since _A_, _P_, _B_, and the point at infinity on _AB_ are four harmonic points, we have the theorem

_Conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes._

*115.* The chord _A"B"_, parallel to the diameter _OQ_, is bisected at _P'_ by the conjugate diameter _OP_. If the chord _A"B"_ meet the asymptotes in _A'_, _B'_, then _A'_, _P'_, _B'_, and the point at infinity are four harmonic points, and therefore _P'_ is the middle point of _A'B'_. Therefore _A'A" = B'B"_ and we have the theorem

_The segments cut off on any chord between the hyperbola and its asymptotes are equal._

*116.* This theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given.

[Figure 29]

FIG. 29

*117.* For the circ.u.mscribed quadrilateral, Brianchon's theorem gave (-- 88) _The lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point._ Take now for two of the tangents the asymptotes, and let _AB_ and _CD_ be any other two (Fig.

29). If _B_ and _D_ are opposite vertices, and also _A_ and _C_, then _AC_ and _BD_ are parallel, and parallel to _PQ_, the line joining the points of contact of _AB_ and _CD_, for these are three of the four lines of the theorem just quoted. The fourth is the line at infinity which joins the point of contact of the asymptotes. It is thus seen that the triangles _ABC_ and _ADC_ are equivalent, and therefore the triangles _AOB_ and _COD_ are also. The tangent AB may be fixed, and the tangent _CD_ chosen arbitrarily; therefore

_The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area._

*118. Equation of hyperbola referred to the asymptotes.* Draw through the point of contact _P_ of the tangent _AB_ two lines, one parallel to one asymptote and the other parallel to the other. One of these lines meets _OB_ at a distance _y_ from _O_, and the other meets _OA_ at a distance _x_ from _O_. Then, since _P_ is the middle point of _AB_, _x_ is one half of _OA_ and _y_ is one half of _OB_. The area of the parallelogram whose adjacent sides are _x_ and _y_ is one half the area of the triangle _AOB_, and therefore, by the preceding paragraph, is constant. This area is equal to _xy __sin__ a_, where a is the constant angle between the asymptotes.

It follows that the product _xy_ is constant, and since _x_ and _y_ are the oblique coordinates of the point _P_, the asymptotes being the axes of reference, we have

_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 a.n.a.lytic 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 circ.u.mscribed 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_ pa.s.ses 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 circ.u.mscribed 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__;_