Hence the angles which OP and OP' subtend either at S or S' are supplementary.

In a similar manner if P and P' are on the same branch of the hyperbola, the angles subtended either at S or S' may be shown to be equal.

PROP. XV.

55. If the tangent at any point P of an hyperbola meet the conjugate axis in the point t, and Pn be drawn at right angles to CB; then

Cn. Ct BC2.

=

.. Ct. Cn: PN:: CT. CN: CN. NT;

56. The proofs that we have given up to this point of the properties of the hyperbola are closely analogous to the corresponding propositions in the ellipse. The remaining properties of the hyperbola are more conveniently investigated by means of its relation to certain lines, which we shall presently define, called Asymptotes, in the same manner as many of the properties of the ellipse were deduced from those of the auxiliary circle.

DEF. The hyperbola described (see fig. Prop. XVI.) with C as centre, and BB' as transverse axis, and AA' as conjugate axis, is called the Conjugate Hyperbola. Its foci, which will be on the line BCB', will evidently be at the same distance from C as those of the original hyperbola,

since

CS2 CA2 + CB".

=

PROP. XVI.

If through any point R on either of the diagonals of the rectangle formed by drawing tangents to the hyperbola and its conjugate at the vertices A, A, B, B', two ordinates RPN, RDM, be drawn at right angles to AA' and BB', and meeting either the hyperbola or its conjugate in the points P and D; then

Let R be a point on the diagonal O'CO; then

RN2: CN :: A02: A C2,

and DM2

CM2 - CB2 :: AC: BC; (Prop. X.)

... RM2 - DM2: BC2: AC2: BC2;

In exactly the same manner, if NR had been produced to meet the conjugate hyperbola in P, and MR had been produced to meet the original hyperbola in D, we should have had,

COR. If RP be produced to meet the hyperbola in p, and the other asymptote in r; then

RN2 – PN2 = RP. Pr; (Euclid, II. 5,)

.. RP. Pr= BC

Hence as RPN is further removed from A, and the line Pr consequently increases, since the rectangle contained by RP and Pr remains constant, RP must diminish, and by taking R sufficiently far from C, RP may be made less than any assignable magnitude. The line CR, therefore, continually approaches nearer and nearer to the hyperbola, though it never actually reaches it.

56. The proofs that we properties of the hyper corresponding propositio properties of the hyperl by means of its rela presently define, calle many of the properti of the auxiliary circl DEF. The hyperl C as centre, and J jugate axis, is c

CR is called an Asymptote to

where NR produced meets the

Shall have
XP.Pr=BC2;

also the asymptote to the conjugate

it may be shown that the other

which will be or the rectangle 00' is an asymptote to both

be the point where the asymptote meets the direc

For by similar triangles

CE

CO: CX: CA,

:: CA: CS, (Prop. II.)

But CO2 CA+ CB = CS2;

=

.. the angle CES is a right angle. (Euclid, VI. 8, Cor.)

PROP. XVIII.

If from any point R in one of the asymptotes to an hyperbola ordinates RPN, RDM be drawn to the hyperbola and its conjugate respectively, and PD be joined, PD will be parallel to the other asymptote.

For RN RM2 :: BC2 : AC2;

and RN - PN: RM2 - DM :: BC: AC, (Prop. XVI.) .. PN2: DM2: BC2: AC2;

:: RN2: RM2,

.. PN : DM :: RN: RM;

.. PD is parallel to MN.

(Euclid, VI. 2.)

Also CN CM:: AC: BC,

.. MN is parallel to AB;
and OA: Ao :: OB: Bo',
.. AB is parallel to oo'.

Hence PD is parallel to oo'.

COR. So also if P and D be the points where NR and MR produced meet respectively the conjugate and the original hyperbola, PD will be still parallel to oo'.