PROP. XIV.

The area of all the parallelograms, circumscribing an ellipse, formed by drawing tangents at the extremities of two conjugate diameters, is constant, each being equal to the rectangle under the axes.

PROP. XV.

The sum of the squares of any two conjugate diameters, is equal to the same. constant quantity, namely, the sum of the squares of the two axes.

The rectangle under the focal distances of any point is equal to the square of

the semi-conjugate.

That is, if CD be conjugate to CP,

SP. HP CD 2.

Draw SY, HZ, perpendiculars to the tangent at P, and PF perpendicular on CD.

D

E

a

H

If two tangents be drawn, one at the principal vertex, the other at the vertex of any other diameter, each meeting the other diameter produced, the two tangential triangles thus formed, will be equal.

That is,

A CPT A CAK.

Draw the ordinate PM, then,

CM: CA: CP: CK,by similar As

But, CM: CA:: CA: CT, Prop. x.

.. CA: CT:: CP: CK.

The two triangles CPT, CAK, have thus the angle C common, and the sides about that angle reciprocally proportional; these triangles are therefore equal.

K

C M

A

Cor. 1. From each of the equal triangles CPT, CAK, take the common space CAOP; there remains,

triangle OAT = triangle OKP.

Cor. 2. Also from the equal triangles CPT, CAK, take the common triangle CPM; there remains,

.. trap. AKPM trap. AKXG :: CA2-CM2: CA2-CG", dividendo.

Cor. 1. The three spaces AKXG, TPXG, GQE, are all equal.

Cor. 2. From the equals AKXG, EQG, take the equals AKRr, Eqr; there remains,

RrXG = rqQG.

Cor. 3. From the equals RrXG, rqQG, take the common space rqvXG there remains,

triangle vQX = triangle vqR.

Cor. 4. From the equals EQG, TPXG, take the common space EvXG; there remains,

TPvE = triangle vQX.

Cor. 5. If we take the particular case in which QG coincides with the minor axis,

The triangle EQG becomes the triangle IBC, The figure AKXG becomes the triangle AKC, .. triangle IBC

= triangle AKC
= triangle CPT.

P

Cor. Hence, any diameter divides the ellipse into two equal parts.

PROP. XX.

The square of the semiordinate to any diameter, is to the rectangle under the abscissæ, as the square of the semi-conjugate to the square of the semi-diameter.

That is,

If Qq be an ordinate to any diameter CP,

Qu2 Pv. vp :: CD: CP2

Produce Qq to meet the major axis in E; Draw QX, DW, perpendicular to the major axis, and meeting PC in X and W.

Then, since the triangles CPT, CvE, are similar,

trian. CPT: trian. CvE :: CP2 : Cv2 or, trian. CPT: trap.TPvE :: CP2 : CP2 — Cv2

:

D

Again, since the triangles CDW, vQX, are similar,
triangle CDW
triangle CDW =
triangle vQX =

But

And

triangle vQX :: triangle CPT;

CD2 : vQ2

Prop. 18., Cor. 5.

trapez. TPvE;

Prop. 18., Cor. 3.

CD2 :: CP2 - Cv2: vQ2

Cor. 1. The squares of the ordinates to any diameter, are to each other as the rectangles under their respective abscissæ.

Cor. 2. The above proposition is merely an extension of the property already proved in Prop. 12, with regard to the relation between ordinates to the axis and their abscissæ.

HYPERBOLA.

DEFINITIONS.

1. AN HYPERBOLA is a plane curve, such that, if from any point in the curve two straight lines be drawn to two given fixed points, the excess of the straight line drawn to one of the points above the other will always be the same.

2. The two given fixed points are called the foci.

Thus, let QAq be an hyperbola, S and H the foci.

Take

any number of points in the curve, P1, P2, P3, .............. Join S, P1, H, P1; S, P2, H, P2; S, P3, H, P3;......then, HP2- SP2 = HP SP3 =

HP1- SP1

3

......

If HP1 SP, and SP, HP'

1

......

be always equal to the same constant quantity, the points P1 P2 P3 ..... and P'1, P2, P3, will lie in two opposite and similar hyperbolas QAq, Qaq', which in this case are called opposite hyperbolas.

3. If a straight line be drawn joining the foci, and bisected, the point of bisection is called the centre.

4. The distance from the centre to either focus is called the eccentricity. 5. Any straight line drawn through the centre, and terminated by two opposite hyperbolas, is called a diameter.

6. The points in which any diameter meets the hyperbolas are called the vertices of that diameter.

7. The diameter which passes through the foci is called the axis major, and the points in which it meets the curves the principal vertices.

8. If a straight line be drawn through the centre at right angles to the major axis, and with a principal vertex as centre, and radius equal to the eccentricity, a circle be described, cutting the straight line in two points, the distance between these points is called the axis minor.