73. If from G the foot of the normal at P a perpendicular GK be drawn to either focal distance, then PK will equal half the latus rectum. 74. If PG, PG' be parts of the normal cut off by the axes major PG b2 PG a2 and minor respectively, prove that 75. Draw a tangent at the extremity of the latus rectum of the x2 y2 ellipse whose equation is + =1. 9a2 4a2 76. A tangent at L (see figure) meets any ordinate NP produced in R; show that NR-SP. 77. A tangent at the extremity of the latus rectum intersects the minor axis produced in the circumference of a circle on the major axis. 78. In an ellipse the tangents at the extremities of any focal chord intersect in the directrix and in a perpendicular to the chord from the focus. 79. If a line be drawn through the focus of an ellipse, making an angle with the major axis, and tangents be drawn at the extremities of this line, these tangents will be inclined to one another 2e at an angle 4, such that tan &= sin 0. I -e2 80. The centre of an ellipse coincides with the vertex of a common parabola, and the axis major of the ellipse is perpendicular to the axis of the parabola. Required the proportion of the axes of the ellipse, so that it may cut the parabola at right angles. A tangent to the parabola at the point of its intersection with the ellipse is a normal to the ellipse at the same point. Hence we may show that =√2. α 81. If with the coordinates of any point in an elliptic quadrant as semi-axes, a concentric ellipse be described, the chord of the quadrant of the former will be a tangent to the latter. The equation to the chord is tangent of the latter ellipse property (5). Ꮖ y + =1; and the equation to the α b may be rendered identical with it by 82. If a, ß; a', B' be the coordinates of two points in a diameter of an ellipse, and be subject to the condition a2b2=a2ßß'+b2aa', show that the equations to the tangents at the extremity of this diameter are a BB'. y+b2 √ aa'.x=±a2b2. 83. In a given equilateral parallelogram inscribe an ellipse of given eccentricity. 84. If ABC be a given triangle, show how to find the ellipse which shall have A and B for its foci, and shall touch a circle of given radius whose centre is C. 85. Through any point in the line joining the centre and intersection of the tangents to any two points of an ellipse, two straight lines are drawn respectively parallel to its diameters passing through the points of contact; prove that the triangles formed by these lines and the tangents are equal. 86. Find the equation to the normal at any point of an ellipse, expressed in terms of its inclination to the axis major. 87. If be the angle which the focal distance to any point of an ellipse makes with the tangent, and the angle between the lines. drawn from that point to the extremities of the axis major, then 2 tane tan . 88. In any ellipse prove that GK (see figure) is equal to e. PN. 89. Find expressions for the chords of curvature through the focus and centre of an ellipse. 90. If e, e be the radii of curvature at the extremities of two conjugate diameters of an ellipse, show that g+g' is constant. Also if c, d be the curvatures at two points at which the tangents are at right angles, c3+c is constant. For the former part of this question use formula (13), and for the latter it may be sufficient to remark, that the curvature at any point of a curve varies inversely as the radius of curvature at that point. 91. Find the length of the normal at any point of an ellipse expressed in terms of its inclination to the axis major. 92. 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. 93. Two conjugate diameters of an ellipse include an angle y; show that these diameters are equal to one another when sin γ is the least possible. In this case find the value of y; a being 8 and b, 5. The value of y is given by the equation tan Y 5 = 2 94. If from the extremities of any diameter of an ellipse chords be drawn to any point in the curve, and one of them be parallel to a diameter, the other will be parallel to the conjugate diameter. 95. The diameters which bisect the lines joining the extremities of the axes of an ellipse are equal and conjugate. In this question formula (12) will be found convenient. 96. If CP=a', and CD the semi-conjugate to CP=b' have such a position that <A'CP-a, and DCB=8; show that If x', y' be the coordinates of P, we may show that 97. A triangle is described about an ellipse; prove that the products of the alternate segments of the sides made by the points of contact are equal. The segments of tangents drawn from the same point without the ellipse are proportional to the diameters respectively parallel to them. 98. If a polygon circumscribe an ellipse, the continued products of the alternate segments are equal to one another. The remark made in the last question may here be repeated. 99. If SP, HP be any two focal distances in an ellipse whose vertex is A; and if AQL be drawn cutting SP in Q and bisecting HP in L; show that the locus of Q is an ellipse whose axes are 2a(1-e) and 2b(1—e). If S be taken for the origin of coordinates, x, y the coordinates of 2y I-e 2X Q, x'y' those of P, then we may show that x'=- ; and y'=- ; which results lead at once to the solution. I-e 100. If C be the centre of an ellipse, and in the normal to any point P, PQ be taken equal to the semi-conjugate at P; show that the locus of Q is a circle of which C is the centre. It may be shown that CQ=a-b, a constant quantity, and therefore the radius of a circle about C. 101. If SQ be drawn always bisecting the angle PSC of an ellipse (see figure) and equal to the mean proportional between SC and SP; find the eccentricity of the locus of Q. Either polar or rectangular coordinates may be used; and the result gives the eccentricity required= 2e 102. CP and CD are semi-conjugate diameters of an ellipse, and PF is a perpendicular let fall upon CD or CD produced; determine the locus of the point F. The equation to the locus of F is found by eliminating the coordinates of P from the equations to CD, PF and the ellipse. The result is (x2 + y2 2 2 a2 b2 103. 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. 104. Prove that the locus of the points of bisection of any number of chords of 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. The eccentricity of this locus is the same as that of the original ellipse. 105. If tangents drawn to any two points of an ellipse meet each other; show that their lengths are inversely as the sines of the angles which they make with the lines drawn to either focus. HYPERBOLA. The general formulæ expressing the properties of the hyperbola are similar to those of the el lipse. So that if any N' H C TA SN A result be obtained for the with those for the hyperbola, will 106. The tangent to any point of a hyperbola is produced to meet the asymptotes; show that the triangle cut off is of constant magnitude. The equation to the hyperbola referred to its asymptotes is xy=(a2+b2). Hence the equation to the tangent at the point 4 107. An ellipse and a hyperbola have the same foci; show that these curves will intersect one another at right angles. 108. Express the length of a perpendicular from the centre of a hyperbola upon the tangent at any point in terms of its inclination to the transverse axis. 109. Find the locus of the extremity of a perpendicular from the centre of a hyperbola upon a tangent at any point. The equation to this locus is (x2+y2)2=a2x2—b2y2. 110. Find the locus of the intersection of two tangents to a hyperbola which meet one another at right angles. This locus is a circle whose radius is √a2-b2. III. A pair of conjugate hyperbolas being given, to find their centre. 112. Investigate the polar equation to the hyperbola, the focus being the pole, having given SP-HP=2AC; and draw the asymptote by means of this equation. 113. If 3CA=2CS in a given hyperbola, find the inclination of the asymptote to the transverse axis. 114. Of two conjugate diameters of a hyperbola, one only meets the curve; and if one be drawn through a given point of the curve, find where the other meets the conjugate hyperbola. 115. If a line intersecting a hyperbola in the point P, and its asymptotes in R, r, move parallel to itself, the rectangle RP.Pr is constant. 116. If A and B be the extremities of the axis major of a hyperbola or an ellipse, T the point where the tangent at P meets AB or AB produced, QTR a line perpendicular to AB and meeting AP and BP in Q and R respectively, then QT=RT. 117. Find the latus rectum and eccentricity of the hyperbola which is conjugate to that whose equation is y2=4(x2+a2). The latus rectum = =8a, the eccentricity = √5. 118. If normals be drawn to an ellipse from a given point within it, the points where they cut the curve will all lie in a rectangular hyperbola which passes through the given point and has its asymptotes parallel to the axes of the ellipse. 119. Find the radius of curvature to a hyperbola at the extremity of its latus rectum, the axes being 20 and 12. The radius of curvature at any point = (normal)3 |