298 the intersections of each pair of hyperbolas lie on the axis of the third. 27. The necessary and sufficient condition in order that the equation la+mẞ2+ny2=0 may represent a rectangular hyperbola is 1+m+ n = 0. 28. Shew that √(la) + √(mß) + √√(ny) = 0 represents in general an ellipse, parabola, or hyperbola according as is positive, zero, or negative; where a, b, c denote the lengths of the sides of the triangle formed by a = 0, ẞ= 0, y = 0. 29. Shew that lẞy + mya+naß = 0 represents in general an ellipse, parabola, or hyperbola according as l2a2 + m2b2 + n2c2 — 2lmab — 2mnbc — 2nlca is negative, zero, or positive. 30. Find the condition that the line λu + μv + vw = 0 lu2 + mv2 + nw2 = 0. may touch the conic section 31. Find the fourth point of intersection of the conic sections and lvw + mwu + nuv = 0, l'vw + m'wu + n'uv = 0. 32. Shew that the equation to the radical axis of the circles inscribed in a triangle and circumscribed about it is 33. Find the equation to the diameter of the curve lBy + mya + naß = 0 which passes through the point of intersection of the lines B=0 and y = 0. 34. Find the equation to the tangent to the curve √ (la) +√(mß) + 1/ (ny) = 0, which is parallel to the line y=0; and thence shew that the centre of the curve is determined by 35. From a point P two tangents are drawn to a conic section meeting it in the points M and N respectively; the line through P which bisects the angle MPN meets the chord MN in Q; any chord of the conic section is drawn through Q; shew that the segments into which the chord is divided by the point Q subtend equal angles at P. SECTIONS OF A CONE. CHAPTER XVI. ANHARMONIC RATIO AND HARMONIC PENCIL. Sections of a Cone. 324. WE shall now shew that the curves which are included under the name conic sections, can be obtained by the intersection of a cone and a plane. DEF. A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which remains fixed. The fixed side is called the axis of the cone. Let OH be the fixed side, and OHC the right-angled triangle which revolves round OH. In order to obtain a cone such as is considered in ordinary synthetical geometry, we should take only a finite line OC; but in analytical geometry it is usual to suppose OC indefinitely produced both ways. A section of the cone made by a plane through OH and OC will meet the cone in a line ÒB, which is the position OC would occupy after revolving half way round. Let a section of the cone be made by a plane perpendicular to the plane BOC; let AP be the section, A being the point where the cutting plane meets OC; we have to find the nature of this curve AP. Let a plane pass through any point P of the curve, and be perpendicular to the axis OH; this plane will obviously meet the cone in a circle DPE, having its diameter DE in the plane BOC. Let MP be the line in which the plane of this circle meets the plane section we are considering, M being in the line DE. Since each of the planes which intersect in MP is perpendicular to the plane BOC, MP is perpendicular to that plane, and therefore to every line in that plane. Draw AF parallel to ED, and ML parallel to OB; join AM. Let AM=x, MP=y, OA=c, HOC=a, OAM = 0; the angle AML will be equal to the inclination of AM to OB, that is, to π Now 0-2α. EM-FL-FA-AL- 2c sin a- AL; AL sin AML sin (π-0-2a) sin + a ; But, from a property of the circle, MP2 = EM . MD ; If we compare this equation with that in Art. 282, we see that the section is an ellipse, hyperbola, or parabola, accordsin sin (+2a) is negative, positive, or zero, that ing as cos2 α is, according as 0+2a is less that π, greater than π, or equal to π. Hence if AM is parallel to OB the section is a parabola, if AM produced through M meets OB the section is an ellipse, if AM produced through A meets OB produced through Othe section is an hyperbola. = If c 0 the section is a point if 0+2a is less than π, two straight lines if 0+2a is greater than 7, and one straight line if 0+2α=π. The section is also a straight line whatever c if 00 or π. may be, The equation above obtained may be written y2 sin 0 sin (0+ 2a) (2c sin a cos a cos2 α sin (0+2α) suppose 0+2a to be less than T, so that the curve is an ellipse; then by comparing this equation with the equation (2ax-x), we have a2 Also e1 b2 cos2 a- {sin3 (0+a) - sin' a} _ cos' (0+ a) cos2 a = cos2 a If we suppose in the figure of Art. 324 that AM is produced to meet the cone again in A', then 2a = AA', as might have been anticipated; also b may be shewn to be a mean proportional between the perpendiculars from A and A′ on the axis OH. Similar results may be obtained when the curve is an hyperbola. |