Similarly

qp.pQ = CD2 = QP. Pq;

or, if Vbe the middle point of Q,

QV2 - PV2 = QV2 − pV2.

COR. If a straight line PP, p.P meet the hyperbola in P, p, and the conjugate hyperbola in P1, P, PP, PP1

For if the line meet the asymptote in Q, q,

QP=Pig and PQ = qp, .. PP1 = PP1•

PROP. 2. A diameter bisects all chords parallel to the tangents at its extremities, i. e. all chords parallel to its conjugate.

This can be proved exactly as in the analogous proposition for the ellipse.

Let QQ, (fig. 91) be any chord of an hyperbola meeting the directrix in R, and let O be the centre point of QQ, and F the focus.

Join FQ, FQ1, and draw FY perpendicular to QQ

but since Q and Q, are on the hyperbola,

FQ: FQ, :: QR : Q ̧R;

where AW is drawn through the vertex parallel to QR meeting the directrix in W.

I.e. OY OR in a constant ratio.

:

1

Take any second chord qq, parallel to QQ, meeting FY in Y, and the directrix in R1. Let O, be its centre point; then, since OY_O,Y, it follows that the line 00, must pass through the OR OR1

1

point T in which FY meets the directrix, and is therefore fixed for all chords parallel to QQ,. This line will evidently pass through the centre (i. e. will be a diameter), for by the last proposition it bisects all chords of the conjugate hyperbola parallel to QQ,, i. e. it bisects the diameter Dd, which is also bisected by C.

Let TO meet the hyperbola in P and suppose qq, to move parallel to itself till it approaches and ultimately coincides with P. Since 0,q=0.9, throughout the motion, the points q, q, will evidently

approach P simultaneously, and in the limiting position qq, will be the tangent at P. It follows that if P, be the other extremity of the diameter through P, the tangent at P, is parallel to QQ,, and therefore to the tangent at P.

COROLLARY 1. The perpendicular on the tangent at any point from the focus meets the corresponding diameter in the directrix.

COR. 2. If the tangent at P meet the asymptotes in E and e, PE Pe, for by the last proposition the intercept between 9 and զ one asymptote is always equal to the intercept between q, and the other asymptote, and when q and q, ultimately coincide with P these intercepts become PE and Pe respectively, i.e. the portion of any tangent between the asymptotes is bisected at the point of

contact.

COR. 3. If PE be the tangent at P meeting the asymptote in E, PE2 = CD2, where CD is the semi-diameter conjugate to CP. For taking a parallel chord very near the tangent meeting the curve in p, p, and the asymptote in e, we have, by Prop. 1,

1

COR. 4. The asymptotes are the diagonals of the parallelogram formed by the tangents at the extremities of a pair of conjugate diameters. For E and e, which are on the asymptotes, are also angular points of such a parallelogram.

PROP. 3. Tangents drawn at the extremities of any chord subtend equal angles at the focus.

Let PQ (fig. 92) be any chord of an hyperbola and let the tangents at P and Q meet in R. Let F be the focus, and from R draw RN, RM perpendicular respectively to FP, FQ; draw RW perpendicular to the directrix and let the tangent at P meet the directrix in E.

Then EF is perpendicular to FP (p. 157), and therefore parallel to RN.

where PK is the perpendicular from P on the directrix.

Hence in the right-angled triangles RFN, RFM, FN=FM, and FR is common.

Therefore the two triangles are equal in all respects, i. e. the angle RFP the angle RFQ, and RN RM.

=

=

PROP. 4. If PCP, be a diameter and QVQ, a chord parallel to the tangent at P and meeting PP, produced in V, and if the tangent at Q meet PP, in T, then CV.CT-CP (Fig. 92).

Let TQ meet the tangents at P and P, in R and r, and F being a focus draw RN perpendicular to the focal distance FP meeting

it in N, rn perpendicular to FP, meeting it in n, and RM, rm perpendicular to the focal distance FQ. Let F be the other focus, and join F,P, F,P,.

=

Since CF CF1, CP=CP1, and the angle FCP = the angle FCP1, therefore the triangles FCP, F,CP, are equal in all respects; and therefore the angle CPF = the angle CP ̧F ̧.

Similarly the angle CPF, the angle CP,F.

=

Therefore the whole angle FPF, the whole angle F,P,F; but the tangents bisect the angles between the focal distances, therefore the angle FPR = the angle FP,r; i.e. the right-angled triangles RPN, rP ̧n are similar, and therefore

RP: rP, :: RN: rn;

but RN = RM and rn = rm (Prop. 3), therefore

COR. 1. Since CV and CP are the same for the point Q1, the tangent at Q, passes through 7, or the tangents at the extremities of any chord intersect on the diameter which bisects that chord.

PROP. 5. If PCP,, DCD, be conjugate diameters, and QV be drawn parallel to CD meeting the hyperbola in Q and CP in V, then QV: PV. PV :: CD2 : CP2.

Let the tangent at Q (fig. 92) meet CP and CD in T and t respectively, and draw QU parallel to CP meeting CD in U.

Then CV.CT = CP2 and CU. Ct = CD2 (Prop. 4);

but

therefore

but

CU=QV,

CD: CP:: QV. Ct CV. CT;

Ct QV CT : VT,

... CD2: CP2 :: QV CV. VT,