Mathematical Questions and Solutions Volume 25

Mathematical Questions and Solutions Volume 25

By (author) 

List price: US$9.02

Currently unavailable

Add to wishlist

AbeBooks may have this title (opens in new window).

Try AbeBooks

Description

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. 1876 edition. Excerpt: ... radius parallel to CA in D, and DE be drawn parallel to OA; also if OB be joined, and OQ be drawn perpendicular thereto; prove that the tangent BP, and the lines DE, OQ all pass through the same point. 4772. (By Prof. Wolstenholme.)--A circle through the foci B, C of a rectangular hyperbola meets the curve in the point Q', the tangent at Q' to the circle meets BC in O, and OL is another tangent to the circle: prove that (1) the locus of L is the (Bernoulli's) Lemniscate whose foci are B, C, and that OQ' is parallel to the straight line bisecting the angle BLC; (2) if OPQ be drawn at right angles to OQ' meeting the circle in P, Q, the locus of P will be a circular cubic of which B, C are two foci, and a third A divides BC in the ratio--1: + 1 (B being the nearer point to O); also (3) (v/2-l)CP-(v/2 + l)BP = 2AP, or AB. CP + AC. BP = 5 -AP; and (4) the angle QXQ' exceeds the angle Q'LP by a right anjle. Solution by the Proposeb.. In the Solutions to Question 4530 (Reprint, Vol. XXII., pp. 108--111) it has been shown that if A, B, C be three points on a circle, and P a fourth point on the circle, such that PA2 = PB. PC, there are four positions of P: one on the arc CA, (P); one on the arc AB, (Q); and two on the arc BC, (P', Q'). Also the straight lines PQ, P Q' are parallel to the external and internal bisectors of the angle A, and meet BC in the same point as the tangent at A to the circle; and P', Q' will be real only when a-ibc. Hence P', Q' will coincide when a-= ibc; and it is in this case that, investigating the loci of P, Q, Q' when B, C are given points, we obtain the results of the present Question 4772. Using L instead of A for the variable vertex of the triangle, the equation 3369. (Proposed by J. J. Walker, M.A.)--The...show more

Product details

  • Paperback
  • 189 x 246 x 1mm | 64g
  • Rarebooksclub.com
  • United States
  • English
  • black & white illustrations
  • 1236797027
  • 9781236797025