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 . hyperbola, and PF be drawn at right anglesto CD; then PU . PF = 2 CP2. CONIC SECTIONS. 105 Since the triangle PHUis similar to the triangle PFC,.:: CP:PF,..PU, PF=PH: CP, = 2 CD2. {Prop. XXIX.) Prop. XXXI. If PI be the chord of the circle of curvature through thefocus of the hyperbola ; then PI .AG = 2 CD\ Let SP meet CD in E; then, since the triangles PIT!and PEP are similar, .*. PI : PU :: PF :
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 . hyperbola, and PF be drawn at right anglesto CD; then PU . PF = 2 CP2. CONIC SECTIONS. 105 Since the triangle PHUis similar to the triangle PFC,.:: CP:PF,..PU, PF=PH: CP, = 2 CD2. {Prop. XXIX.) Prop. XXXI. If PI be the chord of the circle of curvature through thefocus of the hyperbola ; then PI .AG = 2 CD\ Let SP meet CD in E; then, since the triangles PIT!and PEP are similar, .*. PI : PU :: PF : PE. But PE = AC, {Prop. XII. Cor.).-. PI : PU :: PF : AC,.-. PI . AC = PU . PF, = 2 CD0-. {Prop. XXX.) The point where the circle of curvature intersects thehyperbola may be determined as in the case of the ellipse. Pkop. XXXII. 67. If Pbe any point on the hyperbola, and CD be conju-gate to CP; then SP . SP= CD\ Draw PIP parallel to the asymptote CE meeting thedirectrices in / and /, and CB in U. Let the ordinates, NP, 31D meet the asymptote in R, anddraw P W perpendicular to the directrix ; then by similartriangles, PI : PIV :: CE : CX, :: CA: CX. {P>op. XVII.) 106 CONIC But :: SA : AX,:: GA : SP = PI; so SP = PT,.-. SP. SP=PI. PI, = UP2- UP,= GB2 - CE\= GR2 - CA\ (Prop. Ci22 -0D2 = EM2 - DM2, = GA2, (iVop. XVI.).-. 6iZ2 - 6012 = 2,Hence SP. SP = CD2. PEOBLEMS ON THE HYPEBBOLA. 1. The locus of the centre of a circle touching two givencircles is an hyperbola or ellipse. 2. If on the portion of any tangent intercepted between thetangents at the vertices a circle be described, it will passthrough the foci. 3. In an hyperbola the tangents at the vertices will meetthe asymptotes in the circumference of the circle describedon SS as diameter. 4. If from a point P in an hyperbola PIP be drawn parallelto the transverse axis meeting the asymptotes in / and I,then PI .PI =A G\ 5. If a circle be inscribed in the triangle SPS, the locusof its centre is
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
