67. Tangents are drawn from any point on the conic x2 y3 + 4 to the conic + = 1; prove that the normals at the points of contact meet on the conic a2x2+b3y3 (a2 = 49) * • = 2 68. If ABC be a triangle inscribed in an ellipse such that the tangents at the angular points are parallel to the opposite sides, shew that the normals at A, B, C will meet in some point 0. Shew also that for different positions of the triangle the locus of O will be the ellipse 4a2x2+46y= (a3 - b2)3. 69. If the co-ordinates of the feet of the normals to xy = a from the point (X, Y) be x, Y1; X2 Y2; X3, Yз; X41 Y4 then Y = Y1 + Y ̧ + Y3 + Y1, and X = x ̧ + X2+X ̧+X ̧· 70. The locus of the point of intersection of the normals to a conic at the extremities of a chord which is parallel to a given straight line, is a conic. 71. Any tangent to the hyperbola 4xy = ab meets the x2 y3 ellipse + a2 b2 = 1 in points P, Q; shew that the normals to the ellipse at P and Q meet on a fixed diameter of the ellipse. 72. If four normals be drawn from the point O to the ellipse b2x2 + a2y2 = ab3, and P,, Pa, Pa, P, be the perpendiculars from the centre on the tangents to the ellipse drawn at the feet of these normals, then if 1 1 1 1 1 where c is a constant, the locus of O is a hyperbola. 73. Find the locus of a point when the sum of the squares of the four normals from it to an ellipse is constant. 74. The tangents to an ellipse at the feet of the normals which meet in (f,g) form a quadrilateral such that if (x,y), (x", y') be any pair of opposite vertices a2 = b2 -1, and that the equation of the line joining the middle points of the diagonals of the quadrilateral is fx + gy=0. 75. Tangents are drawn to an ellipse at four points which are such that the normals at those points co-intersect; and four rectangles are constructed each having two adjacent sides along the axes of the ellipse, and one of those tangents for a diagonal. Prove that the distant extremities of the other diagonals lie in one straight line. 76. From a point P normals are drawn to an ellipse meeting it in A, B, C, D. If a conic can be described passing through A, B, C, D and a focus of the ellipse and touching the corresponding directrix, shew that P lies on one of two fixed straight lines. 77. If the normals at A, B, C, D meet in a point 0, then will SA. SB.SC. SD = k2. SO3, where S is a focus. 78. From any point four normals are drawn to a rectangular hyperbola; prove that the sum of the squares on these normals is equal to three times the square of the distance of the point from the centre of the hyperbola. x2 y2 79. A chord is drawn to the ellipse+2=1 meeting the major axis in a point whose distance from the centre is a α a + b b At the extremities of this chord normals are drawn to the ellipse; prove that the locus of their point of intersection is a circle. 80. The product of the four normals drawn to a conic from any point is equal to the continued product of the two tangents drawn from that point and of the distances of the point from the asymptotes. 81. Find the equation of the conic to which the straight lines (x + y) - p2 = 0, and (x + μy)3 — q2 = 0 are tangents at the ends of conjugate diameters. 82. From any point 7 on the circle x+y=c2, tangents TP, T'Q are drawn to the cllipse + 1, and the circle TPQ cuts the ellipse again in P'Q'. always touches the ellipse 83. Shew that the line P'Q' b* ̄ ̄ (a2 — b2)2 * A focal chord of a conic cuts the tangents at the ends of the major axis in A, B; shew that the circle on AB as diameter has double contact with the conic. 84. ABCD is any rectangle circumscribing an ellipse whose foci are S and H; shew that the circle ABS or ABH is equal to the auxiliary circle. 85. Any circle is described having its centre on the tangent at the vertex of a parabola, and the four common tangents of the circle and the parabola are drawn; shew that the sum of the tangents of the angles these lines make with the axis of the parabola is zero. 86. Tangents to an ellipse are drawn from any point on the auxiliary circle and intersect the directrix in four points: prove that two of these lie on a straight line passing through the centre, and find where the line through the other two points cuts the major axis. 87. If u = 0, v= O be the equation of two central conics, and u, v, the values of u, v at the centres C, C' of these conics respectively, shew that uv vu is the equation of the locus of the intersection of the lines CP, C'P', where P, P' are two points, one on each curve, such that PP' is parallel to CC'. 0 Examine the case where the conics are similar and similarly situated. 88. Two circles have double internal contact with an ellipse and a third circle passes through the four points of contact. If t, t, T be the tangents drawn from any point on the ellipse to these three circles, prove that tt' = T2. 89. Find the general equation of a conic which has double contact with the two circles (x-a)+y=c2, (x—b)2+ y2=d3, and prove that the equation of the locus of the extremity of the latus rectum of a conic which has double contact with the circles (x± a)2 + y2 = c2 is y3 (x2 — a2) (x2 — a2 + c2) = c*x2. ㄓ 90. Shew that the lines lx+my = 1 and l'x+m'y = 1 are conjugate diameters of any conic through the intersections of the two conics whose equations are (I'm' - l'2m) x2 + 2 (1 − l') mm'xy + (m − m') mm'y2 = 2 (lm' — I'm) x, and 91. (m3l' — m'3l) y3 + 2 (m − m') ll'xy + (1 − l′) ll'x2 = 2 (ml' — m'l) y. If through a fixed point chords of an ellipse be drawn, and on these as diameters circles be described, prove that the other chord of intersection of these circles with the ellipse also passes through a fixed point. 92. The angular points of a triangle are joined to two fixed points; shew that the six points, in which the joining lines meet the opposite sides of the triangle, lie on a conic. 93. If three sides of a quadrilateral inscribed in a conic pass through three fixed points in the same straight line, shew that the fourth side will also pass through a fixed point in that straight line. 94. Two chords of a conic PQ, P'Q' intersect in a fixed point, and PP passes through another fixed point. Shew that QQ' also passes through a fixed point, and that PQ', PQ touch a conic having double contact with the given conic. 95. A line parallel to one of the equi-conjugate diameters of an ellipse cuts the tangents at the ends of the major axis in the points P, Q, and the other tangents from P, Q to the ellipse meet in O; shew that the locus of O is a rectangular hyperbola. 96. L, M, N, R are fixed points on a rectangular hyperbola and P any other point on it, PA is perpendicular to LM and meets NR in a, PC is perpendicular to LN and meets MR in c, PB is perpendicular to LR and meets MN in b. that PA. Pa = PB. Pb = PC. Pc. Prove 97. P is any point on a fixed diameter of a parabola. The normals from P meet the curve in A, B, C. The tangents parallel to PA, PB, PC intersect in A', B', C'. Shew that the ratio of the areas of the triangles ABC, A'B'C' is constant. 98. A point P is taken on the diameter AB of a circle whose centre is C. On AP, BP as diameters circles are described: the locus of the centre of a circle which touches these three circles is an ellipse having C for one of its foci. 99. The straight lines from the centre and foci S, S' of a conic to any point intersect the corresponding chord of contact in V, G, G'; prove that the radical axis of the circles described on SG, S'G' as diameters passes through V. 100. If the sides of a triangle ABC meet two given straight lines in a,, a,; b1, b; c1, c, respectively; and if round the quadrilaterals bb,cc, cc,a,a,, aabb, conics be described; the three other common chords of these conics will each pass through an angular point of ABC, and will all meet in a point. CHAPTER XI. SYSTEMS OF CONICS. 204. THE most general equation of a conic, viz. contains the six constants a, h, b, g, f, c. But, since we may multiply or divide the equation by any constant quantity without changing the relation between x and y which it indicates, there are really only five constants which are fixed for any particular conic, viz. the five ratios of the six constants a, h, b, g, f, c to one another. A conic therefore can be made to satisfy five conditions and no more. For example a conic can be made to pass through five given points, or to pass through four given points and to touch a given straight line. The five conditions which the conic has to satisfy give rise to five equations between the constants, and five independent equations are both necessary and sufficient to determine the five ratios. The given equations may however give more than one set of values of the ratios, and therefore more than one conic may satisfy the given conditions; but the number of such conics will be finite if the conditions are really independent. If there are only four (or less than four) conditions given, an infinite number of conics will satisfy them. The five conditions which any conic can satisfy must be such that each gives rise to one relation among the constants; as, for instance, the condition of passing through a given point, or that of touching a given straight line. |