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A Collection of Examples on the Analytic Geometry of Plane Conics

A Collection of Examples on the Analytic Geometry of Plane Conics


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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1884 edition. Excerpt: ...that the line joining their feet is normal to the conic 2 2 W 2 2#2 + b21/2 =----. 168. If $ and T are the invariants of the pencil which, joins any point on the conic a2 b2 to the feet of the four normals drawn to the curve from x, y, show that /S3 (c4-flV-b2iff Tz a2b2c'x2f 169. If from points on the line Ix + my + n = 0 normals are drawn to the conic-+i-i=. show that the poles of the chords joining their extremities form a quadrilateral inscribed in the cubic tfl _ lA + f 1 _?l + n f-+ t=o a2 b2) b2 a2 a2-b2 a2 b2J 170. A conic circumscribes a triangle so that the normals at the vertices pass through a point; to find the locus of the centre of the curve. If 0 is the angle which the chord joining the points a, 3 makes with the diameter bisecting it, it can be easily c2 shown that cot 0 =-r sin (a + 3); hence we see that the condition (Conies, Art. 231, Ex. 10) that the three normals should meet in a point can be written in the form cot 0i + cot 02 + cot 03 = 0, (1) where 0i, 02, 03 are the angles which the sides of the triangle make with the diameters bisecting them. But if a, fi, y are the perpendiculars from a point on the sides of the triangle, we have a 3 sin B-y sin C + a sin (B-C) COt 0! =; --;, 2 a sin B sin V and similar values for 02, 03; hence, from (1), we obtain-at (j32-72) + "As (j2-2) +-A(a2" P2) = 0, (2) smiu 'J sinJ?w smC1' v 7 which represents a cubic passing through the vertices of the triangle, the centroid, and the centres of the circles touching the sides. It may be observed that if conies be inscribed in the triangle, so that the axis major passes through the centroid, the foci will lie on the cubic (2), 171. In the preceding example show that the locus of the points through which the...


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Product Details
  • ISBN-13: 9781230144931
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 38
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1230144935
  • Publisher Date: 13 Sep 2013
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 86 gr


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