A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . ch all the infinite seriesof circles which can be described iii the same manner as theone in the figure by taking different points on the directrix. 32 CONIC SECTIONS. Prop. II. 23. If Cbe the middle point of A A, then CA is a meanproportional between CS and CX, or CS . CX = CA2. {See Jig. Prop. III.)AX Since SA Alternately 8A .. SA + SA orAA .-. ^ or C01 Again, &4 &4 C7X SASA SA : AX. AX : AX, AX + AX :
A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . ch all the infinite seriesof circles which can be described iii the same manner as theone in the figure by taking different points on the directrix. 32 CONIC SECTIONS. Prop. II. 23. If Cbe the middle point of A A, then CA is a meanproportional between CS and CX, or CS . CX = CA2. {See Jig. Prop. III.)AX Since SA Alternately 8A .. SA + SA orAA .-. ^ or C01 Again, &4 &4 C7X SASA SA : AX. AX : AX, AX + AX : AX:XX : AX,SA : AX, SA : ^X (1)* ^X : AX, AX - AX : AXA A : AX. (2) .-. SA - £4or ££ Alternately SS : AA :: SA : AX;or CS : CJ. :: £J. : AX. Hence from (1) and (2) CA : CX :: CS : CJ,.-. CA2 = CX . CS;or CA is a mean proportional between CS and 6yAr. Cor. Since the three lines CS, CA, CX are proportional,therefore, by the definition of duplicate ratio and Euclid,VI. 20 Cor. CS : CX : CS2 : CA2. (3) Prop. III. 24. If P be any point on the ellipse, then SP+ SP = A A. * The results (1), (2), (3) should be remembered, as they willfrequently be referred to. CONIC Since SP Q. PM and SA AX .-. £P PM So £P: PM SA : AX, AA : II, (Prop. II.) ^ : XX, ^ : IT, ,: SP+ SP: PM + PM ::AA: XX. But PJ/4 PM = MM = XX, .-. SP+ SP= AA. Cor. 1. By means of this property the ellipse maybe prac-tically described and the form of the curve determined. Let a string, equal in length to A A, have its ends fastenedto two points S and S ; and let it be kept stretched bymeans of the point of a pencil at P; then since SP + S Pwill be always equal to A A, the point P will trace out theellipse. Cor. 2. The line A A is the longest line that can be drawnin the ellipse. For, if any other line PQ be drawn, then SP + SQ>PQ,and SP+ SQ>PQ,.+ SP+ SQ + SQ>2PQ, ox AA >PQ. 25. Def. If BOB be drawn at right angles to AC A,meeting the ellipse in B and B, it will be seen further on(Prop.
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
