*107. Diameters. Center.* After what has been said in the last chapter
one would naturally expect to get at the metrical properties of the conic
sections by the introduction of the infinite elements in the plane.
Entering into the theory of poles and polars with these elements, we have
the following definitions:
The polar line of an infinitely distant point is called a _diameter_, and
the pole of the infinitely distant line is called the _center_, of the
conic.
*108.* From the harmonic properties of poles and polars,
_The center bisects all chords through it (§ 39)._
_Every diameter passes through the center._
_All chords through the same point at infinity (that is, each of a set of
parallel chords) are bisected by the diameter which is the polar of that
infinitely distant point._
*109. Conjugate diameters.* We have already defined conjugate lines as
lines which pass each through the pole of the other (§ 100).
_Any diameter bisects all chords parallel to its conjugate._
_The tangents at the extremities of any diameter are parallel, and
parallel to the conjugate diameter._
_Diameters parallel to the sides of a circumscribed parallelogram are
conjugate._
All these theorems are easy exercises for the student.
*110. Classification of conics.* Conics are classified according to their
relation to the infinitely distant line. If a conic has two points in
common with the line at infinity, it is called a _hyperbola_; if it has no
point in common with the infinitely distant line, it is called an
_ellipse_; if it is tangent to the line at infinity, it is called a
_parabola_.
*111.* _In a hyperbola the center is outside the curve_ (§ 101), since the
two tangents to the curve at the points where it meets the line at
infinity determine by their intersection the center. As previously noted,
these two tangents are called the _asymptotes_ of the curve. The ellipse
and the parabola have no asymptotes.
*112.* _The center of the parabola is at infinity, and therefore all its
diameters are parallel,_ for the pole of a tangent line is the point of
contact.
_The locus of the middle points of a series of parallel chords in a
parabola is a diameter, and the direction of the line of centers is the
same for all series of parallel chords._
_The center of an ellipse is within the curve._
[Figure 28]
FIG. 28
*113. Theorems concerning asymptotes.* We derived as a consequence of the
theorem of Brianchon (§ 89) the proposition that if a triangle be
circumscribed about a conic, the lines joining the vertices to the points
of contact of the opposite sides all meet in a point. Take, now, for two
of the tangents the asymptotes of a hyperbola, and let any third tangent
cut them in _A_ and _B_ (Fig. 28). If, then, _O_ is the intersection of
the asymptotes,—and therefore the center of the curve,— then the triangle
_OAB_ is circumscribed about the curve. By the theorem just quoted, the
line through _A_ parallel to _OB_, the line through _B_ parallel to _OA_,
and the line _OP_ through the point of contact of the tangent _AB_ all
meet in a point _C_. But _OACB_ is a parallelogram, and _PA = PB_.
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 circumscribed 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__ α_, where α is the constant angle between the asymptotes.
It follows that the product _xy_ is constant, and since _x_ and _y_ are
the oblique coördinates 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 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 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 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 coördinates 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.
PROBLEMS
1. Draw a chord of a given conic which shall be bisected by a given point
_P_.
2. Show that all chords of a given conic that are bisected by a given
chord are tangent to a parabola.
3. Construct a parabola, given two tangents with their points of contact.
4. Construct a parabola, given three points and the direction of the
diameters.
5. A line _u’_ is drawn through the pole _U_ of a line _u_ and at right
angles to _u_. The line _u_ revolves about a point _P_. Show that the line
_u’_ is tangent to a parabola. (The lines _u_ and _u’_ are called normal
conjugates.)
6. Given a circle and its center _O_, to draw a line through a given point
_P_ parallel to a given line _q_. Prove the following construction: Let
_p_ be the polar of _P_, _Q_ the pole of _q_, and _A_ the intersection of
_p_ with _OQ_. The polar of _A_ is the desired line.