CONIC SECTIONS. DEFINITIONS OF CONIC SECTIONS. 235. A Cone is a solid body, terminating in a point, called its vertex, and having a circle for its base, connected to the vertex by a curved surface, which every where coincides with a straight line passing through its vertex, and through any point in the circumference of the base. If a cone be cut by an imaginary plane, the figure of the section so formed acquires its name according to the inclination or direction of the cutting plane. 236. A plane passing through the vertex of a cone, and meeting the plane of the base, is called a directing plane, and the line of common section is called a directing line. 237. If a cone be cut by a plane parallel to the directing plane, the section is denominated a conic section. 238. If the directing line fall without the base of the cone, the section is called an ellipse. 239. If the directing line touch the circumference of the base, the section is called a parabola. 240. If the directing line fall within the base, the section is called an hyperbola. 241. Equal opposite cones are those which have their axes in the same straight line; and, if cut by a plane through their common line of axis, the sides of the section will be two straight lines cutting each other. Hence the two equal and opposite cones join each other at their vertices, and have their vertical angles equal. 242. If the plane which produces the section of an hyperbola be extended so as to cut the opposite cone, the two sections are denominated opposite hyperbolas. 243. If the plane of a conic section be cut perpendicularly by another plane, which passes through the axis of the cone, the line of common section, in the plane of the figure, is called the primary line. 244. A point where the primary line cuts a conic section is called a vertex of that conic section. Hence the ellipse has two vertices, opposite hyperbolas have each one, and the parabola has one. OF THE ELLIPSE. 245. That portion of the primary line terminated at each extremity by the vertices of the curve, is called the axis major, or transverse axis. 246. A straight line, drawn perpendicularly to the axis major, from any point in it, to meet the curve, is called an ordinate. 247 The middle of the axis major is called the centre of the figure. In the figure here annexed, Aa is the axis major, PM an ordinate to it, and the point C, in the middle of Aa, is the centre of the ellipse, AMa. THE ELLIPSE.—THEOREM 1. 248. The squares of the ordinates of the axis are to each other as the rectangles of the segments of the axis, from each ordinate to each of the two vertices of the curve. Let VDF be a plane passing through the axis of the cone, perpendicular to the cutting plane of the section AlMami, and let Aa be their common section, meeting the conic surface in the points A, a; (then Aa will be the axis major,) and let Q1Ri, NMO»?, be sections of the cone, parallel to the base. Then, because the base DGFE is perpendicular to the plane VDF, the three sections, A1Mawi*', Q1Ri, NMOm, are all perpendicular to the plane VDF; therefore their common sections, It, Mm, are perpendicular to the plane VDF, and to the lines Aa, QR, NO; but, because the plane VDF passes through the axis of the cone, it will divide all the circles parallel to the base into two equal parts; therefore QR and NO are the diameters of the circles Q1R*, Nmojw; and, because the chords It, Mm, are perpendiculars to the diameters, QR, NO, they will be bisected; let H be the point of bisection in 1*, and P the point of bisection in Mm. Let CA = Ca = a, CP = ar, VM = y, CH = s, HI = r,PN = *, Po = M,hq=«, and HR = w. ThenAP = CA+CP=a + ar, aP = Ca-CP =a-x, XT . . M .. . (AVN,AHQ.. v{a+x)= t{a-x) I aPO, aHR .. w (a — x) = u (a + z) QIR y* = vw NMO tu=y\ Wherefore, eliminating /, u, v, w, by multiplying the given equations, the result will be rl(a+x)(a—x) = y*(a—%)(a+ss), or, by actual multiplication, 249. Corollary 1.—Hence, every chord perpendicular to the axis major is bisected by the axis major. 250. Corollary 2.—Hence the tangents at the extremity of the axis major are perpendicular to the axis major. DEFINITIONS RELATIVE TO THE ELLIPSE, CONTINUED. 251. A straight line drawn through the centre, perpendicularly to the axis major, and terminated by the curve, is called the axis minor, or conjugate axis. 252. A third proportional to the axis major and minor, is called the parameter, or the latus rectum of the axes. Thus a and b being the semi-transverse and semi-conjugate axes, 2a: 2b:: 2b :p, the parameter; therefore, ap = 2b1, or if f- \p, we shall have af- V, therefore /=? 253. That point in the axis, cut by an ordinate, which is equal to half the parameter, is called the focus. In the figure here annexed, Bb, drawn through C, is the semi-axis minor; and, if Ll be a third proportional to Aa, Bb, then hi is the parameter, and the point F, where it cuts Aa, is the focus. 254. Any line drawn through the centre, and terminated at each extremity by the curve, is called a diameter. 255. A diameter, which is parallel to a tangent at one extremity of another diameter, is called a conjugate diameter to that other diameter. 256. A straight line, parallel to a tangent, at the extremity of any diameter, terminated at one extremity by that diameter and the curve at the other, is called an ordinate to that diameter. 257. The portion of a diameter between the centre and an ordinate, is called the abscissa of that ordinate, or of that diameter. In the figure here annexed, the straight line, Aa, drawn through the centre, C, is a diameter; and, if ST be a tangent at A, and the diameter Bb be drawn parallel to ST, the diameter is called the conjugate diameter of Aa; and PM, parallel to ST or Bb, is an ordinate to the diameter Aa; and the distance CP, on the diameter Aa, is called the abscissa. ELLIPSE.—THEOREM 2. 258. The square of the axis major is to that of the axis minor as the rectangle contained by the two parts of the axis major, from the ordinate to each vertex, to the square of the ordinate. From the preceding theorem, y2(a2-a^)=y*(a*—**). Now let b represent the semi-axis minor, and let the ordinate y become b, then will its abscissa, z, become zero, and, consequently, y*(a2—a^) =y2(a2—**) will become b2{a2—a?) = a*y2; whence a2: 259. Corollary 1.—Hence every ellipse has two focii at an equal distance from the centre; because y1 = ^ (a2—a?). 260. Corollary 2.—Hence the tangent at either vertex of the curve is parallel to the ordinates; and, consequently, perpendicular to the axis major. ELLIPSE. THEOREM 3. 261. The square of the axis minor is to that of the axis major as the rectangle contained by the two parts of the axis minor, from the ordinate to the extremity of the axis minor, to the square of the ordinate. For, by theorem 2, «2y2 = ft2 (a* - a?) and, by transposition,. ... tfx* = a2 (ft2 — y2) 262. Corollary 1.—Hence the tangent at each extremity of the axis minor is parallel to the ordinates; and, consequently, parallel to the axis major. 263. Corollary 2.—Hence the axis major and minor are reciprocally conjugate diameters. 264. Corollary 3.—If a circle be described on either axis of an ellipse, an ordinate of the circle will be to the corresponding ordinate of the ellipse as the axis of this ordinate is to the other axis. Let PM cut the inscribed circle in N, and let PM be produced to cut the circumscribing circle in N; in both cases let CP=x, PM = y, PN = y, CA = a, and CB=i. By theorem 2, (258,) $»(«*-»*) Therefore, multiplying these two equations, we have 6V:=aV> and extracting the root by=ay; wherefore a : b :: y : y. |