ISBN: 1148935517 / ISBN-13: 9781148935515
A Treatise on the Analytical Geometry of the Point, Line, Circle and Conic Sections: Containing an Account of Its Most Recent Extensions
by John Casey
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from ... Show synopsis 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. 1893 Excerpt: ...such that the normal at P passes through the pole of the normal at Q, prove 4o4 sin a sin a' + 4J cos a cos a' = c4 sin 2a sin 2a'. 60. If three points on an equilateral hyperbola be concyclic with tie centre, the angular points of the triangle formed by tangents at these pointi are concyclic with the centre. 61. The summits of a self-conjugate triangle of an equilateral hyperbola are concyclic with the centre. 62. P, Q are points on an equilateral hyperbola, such that the osculating circle at P passes through Q; the locus of the pole of PQ is (a + = 4y. 53. In the same case the envelope of PQ is 4 (4i--xy)1 = 27/t2 (z8 + y')'. (737) -- a2 _ a 64. The hyperbola-----=-=----cuts orthogonally all the conies a2 4-o3 + P passing through the extremities of the axes of the ellipse--+ = 1. (CBorrox.) a o' i 55. If from any point in the hyperbola-y! = a1 + S a pair of tans' y gents be drawn to the hyperbola----= 1, prove that the four pointi a 4 where they cut the axes are concyclic. 66. If through the point a on an ellipse a line be drawn bisecting the angle formed by the joins of a to the point (a + $), (a-$), prove, if o be constant and 0 variable, that the locus of its intersection with the join of the points (a + /3), (a-fi) is a hyperbola. CHAPTER VIII. MISCELLANEOUS INVESTIGATIONS. Section I.--Figures Inversely Similar. 205. Def.--If upon two given lines AB, A'B' he constructed pairs of similar triangles (ABC, A'B'C), (ABB, A'B'V), -e., such that the directions of rotation ABC and A'B'C, Sfc, are inverse. The two figures ABCD A'B'C If... thus obtained are said to be inversely similar. 206. Double Point And Double Lines. There exists a point S which is its own homologue. This is called the double point, or the centre of similitude. There e...