So then if we draw the hyperbolic section in the plane of the paper and draw through X, X' lines XZ, X'Z' at right

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angles to XX' and PM, PM' perpendiculars to these lines, we have for any point P of the hyperbola

SP: NX (or PM)

:: AS: AX:: A' H : A' X'

HP: NX' (or PM')

ratios of greater inequality.

4. The position of the foci is determined relatively to the magnitude of the axes by the following relation CS CA+ CB3.

=

We have shewn that by giving the proper angle to the cone any hyperbola of given axes may be cut from it by a plane parallel to the axis. Let then the given hyperbola be made by the intersection of a plane parallel to the axis of the cone. The foci will still be the points where the inscribed spheres touch the cutting plane, and these spheres

will be equal. The figure represents the section of the cone and spheres by the plane of the paper.

K'

CV will now be perpendicular to the axis and equal to CB. Also AA' = KK' = 2KV since the circles are equal: .. CA = KV.

Hence CS CA+AS = KV+AK= AV,

.. CS2 = AV2 '= CA2 + CV2 = CA2 + CB2.

5. CX is a third proportional to CS and CA. For in the last figure KX is parallel to CV,

. AV AC :: KV XC,

or CS CA :: CA : CX.

:

=

COR. Each of these ratios AK: AX or AS: AX, the eccentricity.

6. Properties of the secant and tangent.

Art. 6. of Chap. V. applies verbatim to all the Conic Sections.

7. The focal distances make equal angles with the tangent at any point.

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Let the tangent at P meet the directrices in F, F", join SF, HF'; PSF, PHF" will be right angles.

Also the triangles MPF, M' PF" are similar:

and SP: PM: AS: AX :: HP: PM',

.. SP: PH:: PM: PM': PF: PF",

or SP: PF :: HP : PF"'.

2

Hence (Euclid VI. A.) SPF, HPF are similar triangles, and the angles SPF, HPF" are equal.

COR. The tangents at A, A' the vertices of the curve are at right angles to the transverse axes.

8. The feet of the perpendiculars from the foci on any tangent lie on the circle on AA' as diameter.

Let SY, the perpendicular from S on the tangent at P, meet HP in K: join SP, CY. Then in the right-angled triangles SYP, KYP, PY is common and the angle SPY =the angle HPY (by the last proposition).

Hence SY KY, and SP= KP;

=

And C, Y being the middle points of SH, SK, CY is parallel to HK and half of it and therefore = AC.

Hence Y lies on the circle on AA' as diameter. And similarly Z the foot of the perpendicular from H.

COR. 1. CY being parallel to HP and bisecting SH, when produced will also bisect SP. Hence SYP being a right angle, the circle on SP as diameter passes through Y and has its centre on CY produced; hence it touches the auxiliary circle at Y.

COR. 2. If CE be drawn parallel to the tangent at P

and therefore conjugate to CP and intersect PH in E, then CEPY is a parallelogram and PE= CY = AC.

9. If SY, HZ are perpendiculars on the tangent at P from the foci S, H, SY. HZ=BC2.

Since (in the figure of the last Article) YZH is a right angle, YC produced will meet ZH on the circumference of the circle.

The triangles SCY, HCY' are equal in all respects, and SY. HZ= HY'.HZ=HA. HA' = HC2- A' C2 = BC2,

Art. 10 of ChapTV. applies verbatim to the hyperbola.

11. If from any point Q of the tangent PK perpendiculars QN, QT be drawn to the directrix and SP, then

ST: QN AS: AX.

12. Hence if QP, QP' be the tangents to an hyperbola from an exterior point Q, QP, QP subtend equal angles at S.

See Articles 11, 12 of Chap. V.

If P, P' are not on the same branch of the curve, since Q lies on the near side of the directrix to one branch and the further side to the other, Tor T" will lie in PS or P'S produced, and the supplement of one of the angles QSP, QSP' will be equal to the other, i.e. the angles will be supplementary.

COR. Combining this last proposition with that of Art. 7 we see that the point Q is equidistant from the four lines SP, HP, SP, HP'.

13. If QP, QP' be the tangents to an hyperbola from a