PROPOSITION X.

78. In an ellipse the sum of the squares of two conjugate semi-diameters is equal to the sum, and in an hyperbola the difference between the squares of two conjugate semi-diameters is equal to the difference of the squares of the semi-axes of the curve.

Let EC, ZZ' be the axes of the curve, AB and A'B' any two conjugate diameters; it is required to prove that

The ordinates AR, &c. to the transverse axis CE being

drawn; since, in the ellipse, (Fig. 1.) the semi-transverse and semi-conjugate axes being represented by t and c,

substituting in the last equation t2—x2 for x12 (Art. 77.),

c2
t2

0^22==—2 {t2—(t2 —x2)} +t2 —x2.

OA/2

Therefore

OA2 +0A22=ť2+c2. (=0E2+0z2).

In like manner, using the equation

it will be found that in an hyperbola (Fig. 2.)

OA2-OA"2=t2-c2. (=OE2-Oz2).

PROPOSITION XI.

79. In an ellipse and an hyperbola the parallelogram formed on any two semi-conjugate diameters, as sides, is equal to the rectangle formed on the two semi-axes.

For the ellipse, let OE, Oz (Fig. 1.) be the semi-axes, OA and OA', or OB, be two semi-conjugate diameters; also let the angle AOE and EOA' or EOB'=0'.

=

consequently

OA. OA' sin. (0 + 0′ ) =

√ {(t2 sin.3 0 + c2 cos.3 0) (t2 sin.3 0′ + c2 cos.* 0′)}'

and (Pl. Trigon. Art. 75. a.) the first member of this equation is the equivalent of the parallelogram OD.

On multiplying together the terms under the radical sign, the first and last terms of the product, viz.

tsin.20 sin.20' and c4 cos.2 0 cos.20

are (Art. 74.) equal to one another; and therefore their sum may be considered as equal to twice the product of their square roots; that is to

2 t2 c2 sin. sin 'cos. cos. '.

The sum of the other terms of the product under the radical sign is

t2 c2 sin.20 cos.20'+t2c2 sin.20' cos.20,

and the sum just mentioned is evidently equal to twice the product of the square roots of these terms. Thus the whole quantity under the radical sign is the square of the binomial te (sin. cos. '+cos. ◊ sin. 0'),

and the denominator of the fraction is (Pl. Trigon. Art. 31.) equivalent to

The value of OA. OA' sin. (0+0'), that is of the parallelogram OD, is therefore tc, or the rectangle OE.OZ.

For the hyperbola, we have [(c'), Art. 76.], Fig. 2.,

OA. O A' sin. (0′—0)=

member by

✓ {(c2 cos.3 0-t2 sin.o 0) (c2 cos.2 0′ —t2 sin.o

Multiplying both numerator and denominator of the second -1, that member becomes

t2c2 sin. (0-0) √-1

2

{(t2 sin. 20-c2 cos.26) (c2 cos.20'-t2 sin.20')}"

and simplifying the denominator as above, it becomes te (sin. ' cos. -cos. ' sin. ), or te sin. (0'—9). Consequently, OA.OA' sin. (4′ — 0) = tc √ −1.

The imaginary term -1 indicates merely that, in the hyperbola, the parallelogram OD is on the convex sides of the branches of the curve.

PROPOSITION XII.

80. To find, in an ellipse, the positions of two conjugate diameters which are equal to one another.

Let t and c represent the semi-axes of the ellipse, and a each of the conjugate semi-diameters; then (Art. 78.) we t2+c2=2 a2;

have

But it is manifest, from the symmetry of the ellipse with respect to the transverse or conjugate axis, that two equal conjugate diameters must make equal angles with either of those axes; therefore, if represent the angle which each of the equal conjugate diameters makes with the transverse axis, we shall have [(a) Art. 79.]

Ө

Or, since sin. 20-2 sin. cos. (Pl. Trigon. Art. 34.)

Substituting this last value of sin. 20 in the equation for sin. 20, we obtain, after reduction,

Thus the equal conjugate diameters of an ellipse are those which are parallel to lines joining the extremities of the transverse and conjugate axes of the curve.

It is evident from Art. 66. that when two conjugate diameters of an ellipse are equal to one another, the chord which is equal to the parameter of either diameter coincides with the conjugate diameter.

Since t tan. 0=c is the equation for an asymptote of an hyperbola, it is evident that, in the latter curve, the lines which correspond to the equal' conjugate diameters in an ellipse are the asymptotes, or the two infinite diameters.

PROPOSITION XIII.

81. If from one extremity of any diameter of an ellipse or hyperbola a chord be drawn parallel to any other diameter, then a straight line joining the other extremities of the chord and the first diameter will be parallel to a diameter which is conjugate to that other diameter.

In an ellipse (Fig. 1.), let CD, AB be any two diameters, and let CM be a chord parallel to AB; then, if M, D be joined, MD will be parallel to A'B', which is conjugate to AB.

Since CM is, by construction, parallel to A B, it is a double ordinate to A'B', which is by hypothesis conjugate to AB;

therefore CM is bisected by A'B' in R. But the diameter CD is bisected in o; consequently CM and CD are cut proportionally in R and O. Hence it follows (Euc. 2. vI.) that MD is parallel to A'B'.

În a similar manner the proposition may be proved for an hyperbola, (Fig. 2.)

a. DEF. 14. Each of two chords drawn from the extremities of a diameter to the same point in the periphery of an ellipse or hyperbola is called a supplemental chord to the other; and chords thus drawn from the extremities of the transverse axis are called the principal supplemental chords.

PROPOSITION XIV.

82. To determine, in terms of rectangular co-ordinates, the length of a line drawn from one of the foci of an ellipse or hyperbola, or from the focus of a parabola, to any point in the curve.

Let o (Fig. 1.) be the centre, EC the transverse, and zz' the conjugate axis of an ellipse: let also F be a focus, and P any point in the curve, and imagine PR to be drawn perpendicular to E C.

Let OE=t, oz = c, and oF (the excentricity) =e; then the co-ordinate axes, being supposed to coincide with the axes of the curve, we have OR=x, RP=y, and FR=x-e; or if FP is on the other side of the parameter, P'Q', FR= ex: the upper or the lower sign being used according as P is on the right or left of z.