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A Collection of Examples and Problems on Conics; And Some of the Higher Plane Curves

A Collection of Examples and Problems on Conics; And Some of the Higher Plane Curves


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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. 1882 Excerpt: ... tangent drawn from a, /8, 7 to the circle S', hence we have for the equation of the locus sin V(#-a) + sin.B JS-/38) + sin C-/S-7) = 0, which, being satisfied by the line at infinity, represents a curve of the third order. We can obtain the equation of the locus otherwise. For, as can be easily seen, the line joining two poles of a line with respect to a cubic V meets the Hessian in a pair of corresponding points, and the two poles are harmonically conjugate with these points. Hence we have to find the locus of the poles of a line one of whose poles is at infinity. Writing V in the form ax' + fiy + 722 + 8t = 0, where x + y + z + v = Q, and v is the line at infinity, the given cubic (the Hessian of V) is 1-7T-+. I-tt=0, and the locus is ax py yz ov' V/87 (ax'-oV) + V7- W W) + V-/3 (7s2-St?) = 0, which, being divisible by v, represents a curve of the third order. When the given cubic is circular, tbe locus is circular and has two foci in common with the Cayleyan of V; and if it have also its double focus on itself, the locus is a right line. 223. A conic is inscribed in a given triangle; if the foci are conjugate with respect to a fixed equilateral hyperbola, show that they lie on a circular cubic which has its double focus on itself. 224. Show that o(y-), +6(--a?)+o(-y), = 0 represents the nodal tangents, and (25c + ca + ab)x + (2ca + ab + bc)y+ (2ab +bc + ca)z = 0 the line of inflexions of the cubic ax (y--z) + by(z--xf +cz(x--y) = 0. 225. If the tangents at the vertices of a triangle inscribed in a cubic pass through a point, the cubic, referred to the triangle, can be written U= ax (y1-z) + by (z-x) + cz (a?-y2) + 2dzyz = 0, from which it appears that the lines, joining the vertices to the points in which the opposite sides meet the c..


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Product Details
  • ISBN-13: 9781130146448
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 32
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1130146448
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 77 gr


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