SECTIONS OF THE CONE AND GENERAL PROBLEMS. Let AN=x, NP=y, AV=d▲VAN=0, ZAVQ=2a; then will 2d.sin a.sin 0 sin 0.sin (2a+1) y2= cos a is the equation to the conic 120. A conic section is traced upon a plane; draw a tangent from any point without it by means of a ruler and pencil only. 121. If a right cone whose vertical angle is 90° be cut by a plane which is parallel to one touching the slant side, prove that the latus rectum of the section is equal to twice its distance from the vertex. Ꮖ 122. If a right cone of which E the semivertical angle is y be cut Ꮹ P Α' F by a plane making an angle 8 with its axis, the ellipse thus obtained will be such, that the : minor axis major axis = √sin (8+y). sin (d—y): cos y. 123. The section of a right cone by a plane is an ellipse of which a and b are the axes, h and k the distances from the vertex to the points where the plane cuts the sides of the generating triangle; show that a-b2= (h―k)2. If 20 be the vertical angle of cone it may be shown that a2=(h−k)2+4hk. sin2 0, and b2=4hk sin2 0. 124. When the section of a cone is an ellipse or a hyperbola, show that the semi-minor axis is a mean proportional between the perpendiculars dropped from the extremities of the major axis upon the axis of the cone. 125. Find the latus rectum of the section of a right cone made by a plane parallel to the slant side; and when the plane passing through the directrix of the section, and the vertex of the cone is perpendicular to the cutting plane, determine the vertical angle of the cone. The vertical angle is equal to cos-1 126. At what angle must a plane be inclined to the side of a cone in order that the section may be a rectangular hyperbola? Determine also the least vertical angle of the cone for which the problem is possible. 127. Prove that in any conic section the diameter of curvature varies as the cube of the normal. 128. In any conic section the projection of the normal on the focal distance is constant. 129. In any conic section, if S be the focus and G the foot of the normal at P, then SG varies as SP. 130. The sum of the squares of normals at the extremities of conjugate diameters in any conic section is constant. 131. 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 respectively. Show that GK is equal to half the latus rectum. 132. If S be the focus and A the vertex of any conic section, and if LT the tangent at the extremity L of the latus rectum meet AS the axis in T, show that AT equals the eccentricity. 133. 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. 134. If PSP be any straight line drawn through the focus S of a conic section, meeting the curve in the points P and p, and SL be the semi-latus rectum, then will SP, SL and Sp be in harmonical progression. 135. A straight line drawn from the intersection of two tangents to a curve of the second order is harmonically divided by the curve and the chord joining the points of contact. 136. A conic section is cut in four points by a circle, and two lines each passing through two of the points of intersection are made the axes of coordinates, their point of meeting being the origin; show that the equation to this conic section is of the form x2 + bxy + y2+dx+ey+f=0. Under what conditions is the converse true? 137. The distance of a point P from the circumference of a circle its distance from a fixed diameter AB::n:1. Prove that the : locus of P is a conic section. The equation to the locus of P expresses an ellipse or hyperbola, or parabola, according as n is less than, or greater than, or equal to unity respectively. 138. If two lines revolving in the same plane round the points S and H, intersect one another in the point P in such a manner that, (1) SP2+HP2 equals a constant quantity; (2) SP is to HP in the given ratio of n: 1; prove that in each case the locus of P is a circle. 139. Find the locus of the intersection of two normals to a curve of the second order, which cut one another at right angles. If p, be the polar coordinates, the centre being the pole, the re quired locus may be expressed by c= a2-b2 √ a2+b2 This result may be obtained through the medium of rectangular coordinates; and the solution commenced by finding the equation to a normal in terms of its inclination to the transverse or major axis. 140. Find the locus of the intersection of two tangents to a curve of the second order, which cut one another at any given angle. 141. Find the equation of the curve traced out by the extremities of the perpendiculars on the tangents of a circle, drawn from a point in its circumference. 142. If the coordinate axes of the curve whose equation is (x2 — y2)2 = ax(x2 + 3y2) be made to revolve about the origin through an angle of 45°; required the equation to the curve referred to the new axes. 143. Determine the axes to which the rectangular equation a2y2+b2x2=a2b2 must be transferred, so that the transformed equation may be a12+y12=a12. Determine the nature and position of the several curves expressed by the following equations :— 144. 7x+2y+14a=0. 145. 12xy+8x=27y+18. 146. x2+y2-8y+12x+48=0. -- 159. y2—2xy + 3x2 — 2y — 10x+19=0. 160. y2 — 4xy+5x2+2y−4x+2=0. 161. y2−3xy+x2+1=0, the axes being oblique, and inclined to one another at an angle of 60°. 162. y2 — 2xу— x2+2=0. 163. y2x2-y=0. 164. 7y2+16xy + 16x2 + 32y+ 64x + 28=0; referred to the axes used in 161. 165. 4y2-8xy-4x2-4y+28x-15=0. 166. 2y2+x2+4y-2x-6=0. 167. 5y2+6xy+5x2-22y-26x+29=0. 168. y2-2xy+x2-2y+2x=0. 169. y2+2xy+x2+1=0. 170. 5xу+3x2 — 10x+3y−12=0. 171. 7x(y+4)-6y2+8y-3=0. 172. Show that the curve y=5x-7 has the origin of coordi nates for its centre; trace this curve and find the magnitude of its axes. 173. Find the equation to the locus of a point the difference of whose distances from two fixed points is invariable; and trace the curve. 174. The base of a triangle is constant, and the sum of the angles at the base is also constant; find the locus of the vertex. 175. The base of a triangle is given, and one angle at the base is double of the angle at the other extremity of the base; find the locus of the vertex. 176. Determine the locus of a point so situated within a plane triangle, that the sum of the squares of the straight lines drawn from it to the angular points is constant; if the curve has a centre, find its position. 177. Having given the base and area of a triangle, find the locus of the centre of the circumscribed circle. 178. Straight lines are drawn from a fixed point to the several points of a line given in position, and on each as a base is described a triangle whose vertical angle is one-half of each of the angles at the base; find the locus of these vertices. 179. The base of a triangle and the sum of the other two sides are given; find the locus of the centre of the inscribed circle. 180. If two lines SP, HP revolve about the points S, H so that SP x HP CS2 (C being the middle point of SH); it is required to find the locus of P. = 181. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle; to find the locus of their intersection. 182. A straight line revolving in its own plane about a given point intersects a curve line in two points; find the curve when the rectangle of the lines intercepted between the given point and the points of intersection is constant. 183. A straight line of given length 2c is made to move so that its ends are always in contact with two other straight lines which include a given angle 2a; show that the locus of its middle point is an ellipse whose semi-axes are c tan a and c cot a; and the direction of one of its axes bisects the angle included between the two given lines. 184. From two given points A and B two straight lines given in position are drawn; MRQ is a common ordinate to these lines, and MP is taken in MRQ a mean proportional to MQ and MR; required the locus of P. 185. If the base and the difference of the angles at the base of a triangle be given, the locus of the vertex is an equilateral hyperbola. 186. Let AQA' be an ellipse, AA' the axis major, QQ' any double ordinate; join AQ and A'Q' and produce these lines to intersect one another in P; the locus of P is required. 187. To find the locus of the centres of all the circles drawn tangential to a given line, and whose circumferences pass through a given point. 188. Let AQB be a semicircle of which AB is the diameter, BR an indefinite straight line perpendicular to AB, AQR a straight line meeting the circle in Q and BR in R; take AP=QR; required the locus of P. The locus of P is called the Cissoid of Diocles. 189. A point Q is taken in the ordinate MP of the parabola, always equidistant from P and from the vertex of the parabola; required the locus of Q. 190. Let AQB be a semicircle in which AB is the diameter and NQ is any ordinate produced to P, so that NP: NQ=AB : AN; to find the locus of P. The locus of P is called the Witch of Agnesi. 191. Let XX' be an indefinite straight line, A a given point without it, from which draw the straight line ACB perpendicular to XX' which it cuts in the point C, and also any number of straight lines AEP, AE'P', &c.; take EP, always equal to CB; then the locus of P is required. This locus is called the Conchoid of Nicomedes. Trace the curves whose equations are, 192. r = 20 . sin 0 . tan 0. 193. ra2 cos 20. 194. y3 195. xy=±(a+y) ✔/b2—y, when b is greater than a. |