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48. If tangents PT, QT meet in T and one of them QT is produced to any point Q', prove that the angle PTQ' is a mean between PSQ and PHQ.

49. An endless string of greater length than the circumference of an ellipse which is laid on a sheet of paper is made to pass round it and stretched tight by a pencil: prove that the point of the pencil will trace an ellipse having the same foci as the original. (It may be assumed that the normal to the pencil's path at each point will make equal angles with the directions of the string at the point.)

50. If P be the vertex of a triangle whose base AB is bisected in C. Then if AP. BP+ CP2 is a constant quantity, the locus of P is an ellipse.

51. If AD be the portion of the generating line of a cone which contains the vertex of an ellipse cut from it intercepted between the vertex and a line through C at right angles to the axis of the cone; shew that AD=CS.

52. To cut an ellipse of given axes from a given cone.

CHAPTER V.

On the Hyperbola.

1. DEFINITION. Let AVB, BVA' be the two sheets of a cone, whose axis OVO' is in the plane of the paper and AB A'B the two positions of the generating line in that plane. Let the cone be cut by a plane APA' perpendicular to the plane of the paper which cuts it in the line AA', such that A, A' are points in the generating line of the cone on the same side of the axis OVO: then if this plane cuts the cone in the lines PAP PA'p', these lines make up a curve which is called an Hyperbola.

It is manifest that the two branches of this curve may be prolonged to any length and cannot intersect. Also they

B'

B

are each divided into two equal parts by the line AA' produced every line in the plane of the section drawn at right angles to AA' will meet the curve in two points on opposite sides of AA' and at equal distances from it.

AA' is called the transverse axis of the hyperbola, and a line through the middle point of AA', at right angles to AA' in the plane of the section and of a magnitude to be specified in the next article, is called its conjugate axis.

2. Every hyperbola may be projected from or into an hyperbola whose axes are equal, called an equilateral or rectangular hyperbola.

Let DED', RPR' be circular sections of the cone, made by planes at right angles to the axis through C and through

D

P

Nany point of AA' produced, and let this latter plane cut the plane of the curve in PNP at right angles to AA'. Then NP2 = RN. NR'.

Also in the similar triangles NAR, CAD

RN DC AN: AC,

and in the similar triangles NA' R', CA' D'

30

or

NR': CD':: A'N: A'C

:: A'N: AC

.." RN. NR': CD. CD' :: AN. A'N : AC2,

NP: AN.A'N :: CD. CD': AC.

Now if we draw CE from C to touch the circle DED in E, CE=CD. CD': and we may now complete our definition of the conjugate axis of the hyperbola by saying that it is equal in length to CE.

Now let us take a cone whose vertical angle is a right

angle, and let an hyperbola be cut from it by a plane parallel to the axis and such that the transverse axis shall

=

equal AA', then D, D' will merge in V, and CE will = CV, will also CA, since the circle with centre C and distance CA or CA' will pass through V, AVA' being a right angle.

Hence the hyperbola will be rectangular and will have each of its axes equal to AA'.

Let the two hyperbolas be now placed in the same plane with their transverse axes coincident, and let the ordinate NP of the first hyperbola produced if necessary meet the rectangular hyperbola in Q. Then we have

and

hence

or

NP: AN. AN:: CE: AC,

NQ: AN: A'N:: AC2: A C2, or NQAN. A'N:

NP2: NQ2 :: CE2 : AC,

NP: NQ CE : AC.

We shall shew that CE may be greater or less than CA: if CE be greater than CA the first hyperbola will project into the rectangular one if, the transverse axes remaining coincident, they are placed in planes inclined to one another at the proper angle. If CE be less than CA the rectangular hyperbola will project into the one of unequal axes.

COR. The proposition proved above

NP2: AN.A'N:: CE: AC2

might have been equally proved if we had taken the circular section in the other sheet of the cone below A'. Hence we observe that if we take two points in the transverse axis of the curve produced both ways on opposite sides of C and equidistant from it, the ordinates drawn through them will be equal, and the points in which they meet the curve will be equidistant from the centre. Hence the two branches of the

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