27. A pair of conjugate hyperbolas being given, find their centre. 28. Find at what angle a plane must be inclined to the side of a cone in order that the section may be a rectangular hyperbola : and determine the least vertical angle of the cone for which the problem is possible. 29. If two chords of a parabola move parallel to themselves 1828 intersecting each other, the rectangles of their segments are in a constant ratio. 30. In the ellipse all the circumscribing parallelograms are equal. 31. If a right cone of which the semi-angle is y be cut by a plave making an angle d with its axis, the ellipse thus obtained will have its minor-axis : major-axis :: V sin (8 + y) sin (ö - Y): cos y. 32. If a line be drawn through the focus of an ellipse making an angle 0 with the major-axis, and tangents be drawn at the extremities of this line ; these tangents will make an angle , 2e sin 0 such that tan • e2 33. The sum of the squares of any two conjugate diameters in an ellipse is constant. 34. Find the equation to the curve from any point of which if two tangents be drawn to a given ellipse, the angle contained between them shall be constant. 35. If a and b be the semi-axes of an ellipse, and 0 and o the angles which any two conjugates make with the major-axis, 72 prove that tan tan o 36. Having given the equation to an ellipse referred to its principal axes, transform it into one in which the axes are inclined at an angle 0, and in which the axis of y' makes with that of y, a given angle o. Find also the relation between and when the transformed equation is of the same form with the original equation, and shew that in this case each of the new axes is parallel to the tangent drawn at the extremity of the other. q2 1829 t.... the 37. In an ellipse prove that CP? + CD2 AC2 + BC? 2CD2 AC 39. If the distance CP in an ellipse be a mean proportional between the semi-axes, prove that the semi-conjugate CD divides the quadrant of the ellipse into two arcs, whose difference is equal to the difference of the semi-axes. 40. If a straight line DCP be made to revolve about C, and cut the curve PP, P, in as many points as it has dimensions ; 1 1 1 and if be made always equal to CD2 CP2 + CP, locus of the point D will be a conic section whose centre is C. 41. If a line, intersecting an hyperbola in the point P and its asymptotes in R, r, move parallel to itself, the rectangle RP. Pr is constant. 42. The section of a right cone made by a plane will be an ellipse, hyperbola, or parabola. Prove this, and determine the position of the plane for each case. 43. Investigate the formulæ for the transformation of rectangular coordinates to oblique coordinates in the same plane; and hence, having given the equation to a parabola referred to its principal axis, find its equation referred to any other diameter; the abscissa being measured from a point in the curve, and the ordinate being parallel to the tangent at that point. 44. AP is the arc of a conic section, of which the vertex is A; PG the normal, and PK a perpendicular to the chord AP, meet the axis in G and K. Shew that GK is equal to half the latus rectum. 45. The equation to a conic section is 5y + 2xy + 5.x2 – 12x – 12y = 0. Find its centre, and the magnitudes and positions of its principal axes. BC2 46. In the ellipse, if the distances CP, CQ be drawn at right angles to each other, prove that 1 1 1 1 + CP2 CQ ? AC2 47. Investigate the polar equation to the hyperbola, the focus being the pole (given that SP - HP=2AC), and draw the asymptote by means of this equation. 48. If the distance of any point P from a fixed point S be 1830 in a given ratio to its distance PM from a fixed line, find the rectangular equation to the locus of P, according as SP is equal to, less than, or greater than PM. PV.VG CP2 49. In the ellipse, QV2 CD2 50. If CP and CD be semi-conjugate diameters of an hyperbola, prove that CP2 - CD2 = A'C2 BC. 51. In the hyperbola prove that CD.pF = AC.BC. CD2 52. The radius of curvature in the ellipse 1831 = pF 53. Obtain the equation to a straight line which passes through two given points of a parabola, and find what the equation becomes when the points are supposed to coincide. 54. BC, CD are two consecutive arcs of a parabola, the sagittæ of which, bisecting the chords, and parallel to the axis, are equal; prove that the chord of BCD is parallel to the tangent at C. 55. Find the equation to the ellipse, (1) Referred to rectangular coordinates measured from the centre. (2). polar focus. 56. Find the equation to the section of a right cone made by a plane, and determine the position of the plane when the section is a parabola. 57. Determine in what cases the equation Ax2 + By? + Cxy + Dx -+ Ey + F = 0 belongs to each conic section. 58. If S be the focus, and A the vertex of any conic section, and if LT the tangent at the extremity of the latus rectum L AS AT =n:1. 59. The distance of a point p from the circumference of a circle : its distance from a fixed diameter AB Prove that the locus of p is a conic section. 60. In a given equilateral parallelogram inscribe an ellipse of given eccentricity. 61. Prove that the locus of the points of bisection of any number of chords to an ellipse, which pass through the same point, is an ellipse; and find the magnitude and position of the axes when the coordinates to the point are given. 62. Shew that three conditions must be satisfied by the constant coefficients in the general equation to lines of the second order, that it may be the equation to two straight lines. 63. In the parabola, the subtangent to any diameter is double of the abscissa. 1832 64. Prove that the latus rectum of an ellipse or hyperbola 2BC2 AC: 65. CP and CD are semi-conjugate diameters of an ellipse, and PF is a perpendicular let fall upon CD; determine the locus of the point F. 66. In the ellipse the lines SP and HP drawn from the foci S and H make equal angles with the tangent to the ellipse at P. 67. Describe a parabola which shall touch a circle at a given point, and have its axis. coincident with a given diameter of the circle. 68. The section of a right cone by a plane is an ellipse of which a and ß are the axes, a and b the distances from the Vertex to the points where the plane cuts the sides of the generating triangle; shew that a? -- B2 = (a - b)2 69. The equation to the hyperbola being x2 a 62 prove that the equation to its tangent is xx' yy' =l. a2 62 70. Shew that 4y2 + 4x2 + 164 – 8x + 19 = 0 is the equation to a circle, and determine the radius and position of the centre. 71. Find the positions of the asymptotes to the hyperbola ; and the equation to the hyperbola referred to the asymptotes as axes. 72. In the parabola if QQ be a chord drawn parallel to the 1833 tangent at any point P, and PV be drawn parallel to the axis, cutting QQ in V, QQ' will be bisected in V. 73. Find expressions for the chords of curvature through the focus and centre at any point of an ellipse. 74. If a right line be drawn from the extremity of any diameter of an ellipse to the focus, the part intercepted by the conjugate diameter is equal to the semi-axis major. 75. If at the extremities of any two conjugate diameters of an hyperbola, tangents be drawn so as to form a parallelogram, the areas of all such parallelograms are equal. 76. When a conic section is described by the extremity P of a straight line, whose other extremity A and a given point B in it move in straight lines intersecting at right angles in C; prove that if the rectangle ACBD be completed, and PD be joined, PD is a normal at P. 77. Shew that the slant section of a cylinder by a plane is an ellipse; and determine the axes, having given the radius of the base of the cylinder, and the inclination of the cutting plane to its axis. 78. In any conic section if two chords move parallel to themselves and intersect each other, the ratio of the rectangles of their segments is invariable. 79. Investigate the polar equation to a parabola, and apply 1834 it to determine the length of the latus rectum. |