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Conic Sections; Treated Geometrically

Conic Sections; Treated Geometrically


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt: ...sides. 180. Prop. VII. The projection of a parabola is a parabola. For if PNP' be a principal chord, bisected by the axis AN, the projection pnp' will be bisected by the projection an. Moreover pn: PN will be a constant ratio, as also will be an: AN. And PN = 4AS. AN. Hence pn will be to 4tAS. an in a constant ratio, and the projection is a parabola, the tangent at a being parallel to pn. 181. Prop. VIII. An hyperbola can be always projected into a rectangular hyperbola. For the asymptotes can be projected into two straight lines cl, cl' at right angles, and if PM, PN be parallels to the asymptotes from a point P of the curve, PM. PN is constant. But pm: PM and pn: PiVr are constant ratios;.'. pm. pn is constant. And since pm and pn are perpendicular respectively to cl and cl', it follows that the projection is a rectangular hyperbola. The same proof evidently shews that any projection of an hyperbola is also an hyperbola. EXAMPLES. 1. A parallelogram is inscribed in a given ellipse; shew that its sides are parallel to conjugate diameters, and find its greatest area. 2. TP, TQ are tangents to an ellipse, and CP', CQ' are parallel semidiameters; PQ is parallel to P'Q'. 3. If a straight line meet two concentric similar and similarly situated ellipses, the portions intercepted between the curves are equal. 4. Find the locus of the point of intersection of the tangents at the extremities of pairs of conjugate diameters of an ellipse. 5. Find the locus of the middle points of the lines joining the extremities of conjugate diameters. 6. If a tangent be drawn at the extremity of the major axis meeting two equal conjugate diameters CP, CD produced in T and t; then PIP = 2 ATK 7. If a chord AQ drawn from the vertex be produced to meet the minor axis in 0, and CP be ...


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Product Details
  • ISBN-13: 9781231019122
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 68
  • Spine Width: 4 mm
  • Width: 189 mm
  • ISBN-10: 1231019123
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 141 gr

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