95. Let two diameters meet the directrix in p, d. Then if Cd bisects chords parallel to Cp, it may be easily shewn that S is the Orthocentre of the triangle pCd, and hence that Cp bisects chords parallel to Cd. Also that pX. dX: CX2 = CB2 : CA2, which agrees with Art. 70. ¶96. To shew that a conic is concave to its axis. [64, 5. In the figure on p. 25, since SR bisects the angle supplementary to PSQ, therefore if O be any point in the chord PQ, SO: OR < SP : PR. [Introd. But if O be the point, above the axis, in which the ordinate of O cuts the conic, then SƠ : OR = SP: PR. [44. Hence SO', being greater than SO, lies above it; or every arc PQ, however small, is more remote from the axis than the chord of the arc. CHAPTER IV. TANGENT-PROPERTIES OF THE ELLIPSE. 97. THE tangents at the ends of any diameter are parallel to the ordinates of that diameter and to one another. Conversely, parallel tangents touch the curve at opposite ends of a diameter. [17. 98. The tangents at the ends of either axis are parallel to the other. 99. The sides of a parallelogram described as in Art. 77 are tangents. Hence The area of a circumscribing parallelogram which has its sides parallel to conjugate diameters is constant, and equal to the rectangle contained by the axes. 100. Let the normal at P meet DD' in F. Then PF. CD parallelogram PD = СА. СВ. 101. In Art. 86 it is shewn that the rectangles contained by the segments of any two intersecting chords are to one another as the rectangles contained by the segments of any other two chords parallel to the former. Hence if two of the chords, moving parallel to themselves, become tangents TP, Tp, and the other two chords become diameters DD', dd' parallel to those tangents, then the proposition assumes the form TP2 : Tp2 = CD2 : Cď3. Hence any two tangents TP, Tp are as the parallel radii CD, Cd. 102. In Art. 82 suppose the points Q, q to become coincident with P, p. Then the lines PQ, pq become tangents to the ellipse at P and to the auxiliary circle at p respectively, and they always meet in a point T on the axis. Hence, the radius Op being perpendicular to the tangent pT, CN. CT = Cp2 = CA. We proceed to prove a more general theorem which includes this. 103. PROP. XV. If CV be the abscissa of a point Q, the tangent at which meets the radius of abscissæ CP in T, then CV.CTCP2. P R The tangent at P is parallel to QV. Let it meet QT in R. Complete the parallelogram QRPO, viz. by drawing PO parallel to RQ. Then the diagonal RO bisects PQ. But PQ is a chord of contact, viz. of tangents RP, RQ; and therefore its bisector RO is a diameter. Let it be produced to the centre C. Then by parallels CV: CP CO: CR = [22. Therefore CP : CT. CV. CTCP2. 104. If P coincide with A, and N be written for V. then CN. CT CA2. The corresponding property of the minor axis may be expressed, Cn. Ct = CB2. ¶105. PROP. XVI. Tangents at the ends of a focal chord meet on the directrix; and every tangent, measured from the curve to the directrix, subtends a right angle at the focus. R P (i) Let Pp, Qq be any two focal chords; and let PQ meet the directrix in R. Then from the definition, SP: SQ=PR: QR; or R lies on the bisector of <pSQ. [45. [Euc. VI. A. So too it may be shewn that pq meets the directrix on the bisector of 4 PSQ. That is to say, PQ, pq meet the directrix in the same point R. (ii) Now let Pp, Qq be adjacent chords. And let Qq turn about S until it coincides with Pp. Thus the joining lines PQ, pq, which always meet on the directrix, become the tangents at P, p. And since SR is always equally inclined to Sp, SQ, therefore in the limit, when SQ coincides with SP, it makes equal angles with Sp, SP. That is to say, SR is at right angles to Pp, the chord of contact of tangents from R. Compare the next figure. ¶106. Conversely, the chord of contact of the tangents drawn from any point on the directrix passes through the focus. ¶107. DEF. The point of intersection of the tangents at the ends of any chord is said to be the Pole of the chord; and the chord of contact of the tangents from any point is said to be the Polar of the point. ¶108. PROP. XVII. If from any point T on the tangent at P to a conic, perpendiculars TL, TN be let fall on SP and the directrix, then SL: TN=SA : AX. As in Art. 28, SL: SP=RT : RP Alternately, ¶109. COR. Conversely, to draw tangents to a conic. from a given point T. With radius SL determined from the proposition describe a circle about S. Draw tangents TL, TM to the circle. Then SL, SM meet the conic in the required. points of contact. ¶110. PROP. XVIII. The focal radius to the pole T of chord of a conic makes equal angles with the radii to its ends P, Q*. any With the construction of Art. 29, SL: TN=SA : AX, SM: TN=SA : AX, since T lies on the tangent at P. And since T lies on the tangent at Q. Hence, in the right-angled triangles SLT, SMT, the sides SL, SM are equal. And ST is common. And ST is common. Therefore the angles TSL, TSM are equal, or ST makes equal angles with SP, SQ. 111. DEF. The point in which a chord meets the directrix is called the Foot of the chordt. 112. The focal radii to the foot R and the pole T of any chord PQ include a right angle. For SR, ST bisect supplementary angles pSQ, PSQ. [105, 110. * The enunciation of Art. 29 applies in all cases except when the tangents are drawn to opposite branches of a hyperbola. It will appear in the sequel that tangents so drawn subtend supplementary angles at either focus. + This definition is borrowed from Mr H. G. Day's treatise on the Ellipse. |