Geometrical Conics Volume 2 (Paperback)

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. 1894 edition. Excerpt: ...themselves. 171. Pkop. XXIV.--If the tangent at any point P meets the asymptotes in T, t, and if CD is the diameter conjugate to CP, PT = Pt = CD For since the asymptote CT is conjugate to itself, we may consider the tangent PT as meeting a pair of conjugate diameters in two coin-cident points T, .: by Art. 138 Hence the portion of any tangent intercepted between the asymptotes is bisected at the point of contact, and is equal to the conjugate diameter. Cor.--If the portion of a straight line intercepted between the asymptotes is bisected by a point on the curve, the straight line is a tangent Ex. 1.--Given the asymptotes and a point P on the curve, show how (1) to draw the tangent at P, (2) to find the point of contact of a tangent parallel to a given straight line. Ex. 2.--Show that the angle between the asymptotes CT, Cl is equal to, less than, or greater than a right angle, according as CP is equal to, greater than, or less than its conjugate CD. Ex. 3.--If any transverse diameter CP is greater than its conjugate CD, then every transverse diameter is greater than its conjugate; and if equal, equal; and if less, less. Ex. i.--TP, TP are tangents to a hyperbola, and TLM is drawn parallel to an asymptote, meeting PP' in M and the curve in L. Show that Til is bisected by the curve, t Ex. 5.--TP, TP are two tangents, V the mid-point of PP. Through T is drawn TQ parallel to PP, and VQ is drawn parallel to an asymptote. Show that VQ is bisected by the curve.J Ex. 6.--TP is A tangent, and TL is drawn parallel to an asymptote, meeting the curve in L, and is produced to M so that TL = Lil. PM cuts the curve in P. Show that TP is the tangent at P.% Ex. 7.--On the tangent at P are taken points T, T' such that PT = PT' = CD. Show that CT, CT' do...