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 . Join SB, SB; then since SB + SB = A A, {Prop. III.)and that SB = SB,.-. SB = BC2= SB2- CS2,. CA2- GS\ CONIC SECTIONS. 35 Again, SL: SX:: SA : AX, :: CS: GA, [Prop. II.) SX, = CS . CX - CS2, (Euclid, II. 3,)= CA2 - CS2, (Prop. II.)= BC2. Prop. V. 27. The sum of the distances of any point from the fociof an ellipse will be less or greater than A A according asthe point is inside or ou
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 . Join SB, SB; then since SB + SB = A A, {Prop. III.)and that SB = SB,.-. SB = BC2= SB2- CS2,. CA2- GS\ CONIC SECTIONS. 35 Again, SL: SX:: SA : AX, :: CS: GA, [Prop. II.) SX, = CS . CX - CS2, (Euclid, II. 3,)= CA2 - CS2, (Prop. II.)= BC2. Prop. V. 27. The sum of the distances of any point from the fociof an ellipse will be less or greater than A A according asthe point is inside or outside the a S (1) Let Q be a point inside the ellipse. Join S Q, SQ ; and produce S Q to meet the ellipse in P ;join SP; then since SP+ QP> SQ, .-. SP+ SP > SQ+ SP + SP= A A, (Prop. III.).-. SQ+ SQ SP,.-. SQ+ SQ> SP+SP,But SP + SP= A A, (Prop. III.).-. SQ+ SQ> A A. d 2 36 CONIC SECTIONS. Cor. Conversely, a point will be inside or outside theellipse, according as the sum of its distances from the foci isless or greater than A A. 28. Def. If a point P be taken on the ellipse near to P,{see fig. Prop. VI.) and PP be joined, the line PP produced,in the limiting position which it assumes when P is made toapproach indefinitely near to P, is called the Tangent to theellipse at the point P. Peop. VI. If the tangent to the ellipse at any point P intersect thedirectrix in the point Z, and if S be the focus correspondingto the directrix on which Z i
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
