PROP. XXI.

60. If from any point P of a hyperbola, PH and PK be drawn parallel to the asymptotes, meeting them in H and K respectively; then 4. PH.PK = CS2.

Draw the ordinate RPNr meeting the asymptotes in R and r; then by similar triangles,

PH: PR: Co: Oo,

and PK: Pr :: CO: Oo,

:. PH.PK : PR. Pr :: CO2: 002,

:: CS24BC2.

But PR. Pr= BC2,

Ᏼ Ꮯ

.. 4. PH.PK = CS2.

PROP. XXII.

If the tangent at any point P of a hyperbola meet the asymptotes in L and 7; then the area of the triangle L Cl is equal to the rectangle contained by AC and BC.

Draw PH and PK parallel to the asymptotes meeting them in H and K; then

.. CL. Cl 4 PH.PK CS2, (Prop. XXI.)

=

= CO. Co,

=

.. CL: CO :: Co : Cl,

.. the triangles L Cl, O Co have the angle at C common and the sides about those angles reciprocally proportional;

.. the triangle L Cl = the triangle O Co,

= AC.AO,

= AC. BC.

PROP. XXIII.

61. If from any point R in the asymptote of a hyperbola, two ordinates RPN and RDM be drawn to the hyperbola and its conjugate respectively, then the tangents at P and D will be parallel respectively to CD and CP.

Join PD, meeting CR in H; then

since PD is parallel to oo', (Prop. XVIII.)

the tangents at P and D will each meet CR produced in the same point L. (Prop. XXII.)

Produce LP and LD to meet the other asymptotes in 7 and l'; then

since CL. Cl CS' CL. CU, (Prop. XXII.)
= CS2 =

=

...ic: cv:: IP: PL,

.. CP is parallel to the tangent at D.
Also l'D DL :: 'C: Cl,

.. CD is parallel to the tangent at P.

The lines CP and CD are called Conjugate Diameters, since each of these lines is parallel to the tangent at the extremity of the other.

PROP. XXIV.

If CP and CD be semi-conjugate diameters in the hyperbola; then

Draw the ordinates NPR, MDR meeting the asymptote in the point R (Prop. XXIII.); then

H

62. The area of any parallelogram proved by drawing tangents to the hyperbola and its conjugate at the extremities P, P, D, D' of a pair of conjugate diameters is equal to the rectangle contained by the axes.

Let LIL'l' be the parallelogram formed by drawing tangents at the extremities P, P, D, D', of any pair of conjugate diameters. The points L, L', l, l', will (Prop. XXIII.) be on the asymptotes.

Now the parallelogram LL' = 4 parallelogram CL,

63. DEF. The line QV drawn from any point of the hyperbola parallel to the tangent at any point P, and meeting CP produced in V, is called an Ordinate to the diameter CP.

PROP. XXVI.

If QV be an ordinate to the diameter P'CP, and CD be conjugate to CP; then

QV2: PV. P'V:: CD2: CP2.

Produce VQ to meet the asymptotes in R and r; and let the tangent at P meet the asymptotes in L and 7; then

2

RV2 PL :: CV2: CP2,

.. RV2 – PL2 : PL2 :: CV2 – CP2 : CP2.

And CV CP = PV. P' V, (Euclid, II. 6.)

P'V,

.. QV2: PL2 :: PV. P'V: CP2. Alternately, QV2: PV. P'V :: PL2: CP2. But since PD is a parallelogram, (Prop. XXIII.)

.. PL = CD.

Hence QV PV. P'V:: CD2: CP2.