33. If V be the middle point of a chord, the lines bisecting the angles between CV and the chord are parallel to the asymptotes.

34. The lines bisecting the angles between supplemental chords are parallel to the asymptotes.

35. The angle between lines drawn from two points on the curve to one extremity of a diameter equals or is the supplement of that between the lines from the same points to the other extremity.

36. Diameters at right angles are equal to one another.

37. Of two chords of the curve at right angles to one another one has its extremities on the same branch, the other on different branches.

38. If AB, CD are chords at right angles to one another and the circle ABC cuts CD produced in E, BA will bisect ED.

39. The four circles that may be drawn through three of the points A, B, C, D in the last example consist of two pairs of equal circles.

40. If a tangent be at right angles to a chord, the circle on the chord as diameter will pass through the point of contact.

41. The line joining one end of a diameter with one end of a chord at right angles to it is at right angles to the line joining the other end of the diameter with the other end of the chord.

CHAPTER VII.

On the Focal properties of the Hyperbola.

1. LIKE the Ellipse, the Hyperbola has two foci and directrices: before treating of them let us premise the following propositions:

If the circles escribed on the sides VA, VA' of the triangle VAA' touch them and the produced sides in L, S, K,

H

K

L', H, K' respectively, then AS A'H and KL or K'L = AA'.

For SH-SA+ AH = AL+ AK'

=2AL+LK',

SA'+A'H=A'K+A' L'=2A' L' + KL':

also

and LK'

=

KL',

.. AL= A'L' or AS= A'H.

Add AA' to each of these equals,

..AS+ AA = AH = AK' = AL+ LK' = AS+LK';

.. AA' = LK' or KL'.

Also, if we produce KL, K'L' to meet AA' in X, X' (as in the figure of the next article), AX= A′ X'.

For since AL= A' L' and these lines are equally inclined to the line joining the centre of the circles, their projections on this line will be equal: and AX, A' X' have the same projections on the same line as AL, A' L' and therefore have their projections on it equal, and being in the same straight line are themselves equal.

2. Now let AA' be the line in which a plane perpendicular to the plane of the paper cuts it. Let AP, A' P′ be the two branches of the hyperbola in which the same plane cuts the two sheets of the cone AVK, A' VK' whose axis is the line joining the centres of the circles in the last Article: and let the circles SLK, HL' K revolve about the axis, they will form spheres that touch the cone in circles LRK, L'R'K'.

Let P be any point in the hyperbola: the generating line PV of the cone will touch the spheres, in points. R, R' suppose. Join SP, HP; these lines lie in the plane that touches the two spheres at S and H, and therefore touch them in those points: hence SP PR, HP=PR';

=

:. HP SP ᏢᎡ PR

=

=

independent of the position of the point P: hence the differ

=

KĽ' =
RR

ence of the distances of any point in the hyperbola from S and H is constant. S and H are called the foci.

3. Now let the plane through P perpendicular to the axis of the cone cut the cone in the circle QPQ' and the plane of the hyperbola in NP at right angles to A'A produced. Then we shall have SP= PR=QK, and the triangles QAN, KAX are similar;

.. QK : NX :: AK : AX :: AS : AX.

Hence, wherever P is situated on the curve SP: NX is a constant ratio = AS: AX.

Similarly, joining HP,

HP=PR' = QK',

and the triangles QAN, K'AX' are similar;

.. QK': NX' :: AK': AX' :: AK: AX:: AS: AX.

=

Hence also HP: NX' is a constant ratio the same as SP: NX, wherever P is situated on the curve. We have shewn that HA': A' X' SA: AX: these ratios = A'L': A'X' or AK: AX; and the triangles AXK, A' X'L' are both right angled at X or one of them has an obtuse angle at X; in either case we see that the ratios are ratios of greater inequality. As in the ellipse either of these ratios is called the eccentricity.