SEC r. IV. 1056. The conic sections, in geometry, are those lines formed by the intersections of a plans with the surface of a cone, and which assume different forms and acquire different propertics according to the several directions of such plane in respect of the axis of the cone. Their species are five in number. 1057. DEFINItions.—1. A plane passing through the vertex of a cone meeting the plane of the base or of the base produced is called the directing plane. The plane VRX (fig. 404.) is the directing plane, The line in which the directing plane meets the plane of the base or the plane of the base produced is called the direcurir. The line RX is the directrix. 3. If a cone be cut by a plane parallel to the directing plane, the section is called a conic section, as AMIB or AHI (fig. 405.) 4. If the plane of a conic section be cut by another plane at right angles passing along the axis of the cone, the common section of the two planes is called the Fig. 404. rig: ". line of the axis. 5. The point or points in which the line of the axis is cut by the conic surface is or are called the vertex or vertices of the conic section. Thus the points A and B (figs. 404. and 405.) are both vertices, as is the point A or vertex (fig. 406.). 6. If the line of the axis be cut in two points by the conic surface, or by the surfaces of the two opposite cones, the portion of the line thus intercepted is called the primary axis. The line AB (figs. 404. and 405.) and AH (fig. 406.) is called the primary axis. 7. If a straight line be drawn in a conic section perpendicular to the line of the axis so as to meet the curve, such straight line is called an ordinate, as PM in the above figures. The abscissa of an ordinate is that portion of the line of axis contained between the vertex and an ordinate to that line of axis. Thus in figs. 404, 405, and 406. the parts AP, BP of the line of axis are the abscissas AP BP. 9. If the primary axis be bisected, the bisecting point is called the centre of the conic section. 10. If the directrix fall without the base of the cone, the section made by the cutting plane is called an ellipse. Thus, in fig. 404., the section A MB is an ellipse. It is evident that, since the plane of section will cut every straight line drawn from the vertex of the cone to any point in the circumference of the base, every straight line drawn within the figure will be limited by the conic surface. Hence the axis, the ordinates, and abscissas will be terminated by the curve. 11. If the directrix fall within the base of the cone, the section made by the cutting plane is called an hyperbola. Hence it is evident, that since the directing plane passes alike through both cones, the plane of section will cut each of them, and therefore two sections will be formed. And as every straight line on the surface of the cone and on the same side of the directing plane cannot meet the cutting plane, neither figure can be enclosed. 12. If the directrix touch the curve forming the base of the cone, the section made by the cutting plane is a parabola Fig. 106. of The ELLIPSis. 1058. The primary axis of an ellipsis is called the major axis, as A B (fig. 407.); and a straight line DE drawn through its centre perpendicular to it, and terminated at each extremity by the curve, is called the minor axis. 1059. A straight line VQ drawn through the centre and terminated at each extremity by the curve is called a diameter. Hence p the two axes are also diameters. Fig. 407. 1060. The extremities of a diameter which terminate in the curve are called the vertices of that diameter. Thus the points V and Q are the vertices of the diameter VQ. 1061. A straight line drawn from any point of a diameter parallel to a tangent at either extremity of the diameter to meet the curves is called an ordinate to the two abscissas. Thus PM, being parallel to a tangent at V, is an ordinate to the two abscissas VP, PQ. 1062. If a diameter be drawn through the centre parallel to a tangent at the extremity of another diameter, these two diameters are called conjugate diameters. Thus VQ and RS are conjugate diameters. 1063. A third proportional to any diameter and its conjugate is called the parameter or latus rectum. 1064. The points in the axis where the ordinate is equal to the semi-parameter are called the foci. 1065. THEoREM I. In the ellipsis the squares of the ordinates of an axis are to each other as the rectangles of their abscissas. Let AWB (fig. 408.) be a plane passing through the axis of the cone, and Al B another section of the cone perpendicular to the plane of the former; A B the axis of the elliptic section, and PM, HI ordinates perpen- V dicular to it; then it will be their diameters, to which PMI, HI are ordinates as well as to the axis of the ellipse. Now, in the similar triangles APL, AHN, 1067. THEoREM III. In every ellipsis, the square of the minor aris is to the square of the major axis as the difference of the squares of half the minor aris and the distance of an ordinate from the centre on the minor aris to the | ". p Coroll. 1. If a circle be described on each axis as a diameter, one being inscribed within the ellipse, and the other circumscribed about it, then an ordinate in the circle will be to the corresponding ordinate in the ellipsis as the axis belonging to this ordinate is to the axis belonging to the other; that is, CA: CE:: PG : PM, and CE. : CA::pg : pyl ; and since CA2 : CE*:: A P x PB : PM2, >~ and because AP x PB = PG2; CA2 : CE*:: PG2 : PM3, Fig. 411. or CA : CE :: PG : PM. In the same manner it may be shown that CE. : CA::pg : pyl, or, alternately, CA: CE::pM : pg , therefore, by equality, PG : PM::p M : pg, or PG: Cp::CP : pg: therefore CyG is a continued straight line. Coroll. 2. Hence, also, as the ellipsis and circle are made up of the same number of corresponding ordinates, which are all in the same proportion as the two axes, it follows that the area of the whole circle and of the ellipsis, as also of any like parts of them, are in the same ratio, or as the square of the diameter to the rectangle of the two axes; that is, the area of the two circles and of the ellipsis are as the square of each axis and the rectangle of the two; and therefore the ellipsis is a mean proportional between the two circles. Coroll. 3. Draw MQ parallel to GC, meeting ED in Q; then will QM =CG = CA; and let R be the point where QM cuts AB; then, because QMGC is a parallelogram, QM is equal to CG = CE; and therefore, since QM is equal to CA, half the major axis and RM =CE, half the minor axis Q it is the difference of the two semi-axes, and hence we have a method of describing the ellipsis. This is the principle of the trammel, so well known among workmen. If we conceive it to move in the line DE, and the point R in the line AB, while the point M is carried from A, towards E, B, D, until it return to A, the point M will in its progress describe the curve of an ellipsis. 1068. THEoREM IV. The square of the distance of the foci from the centre of an ellipsis is equal to the difference of the square of the semi-ares. Let AB (fig. 4.12.) be the major axis, C the centre, F the focus, and FG the semi-parameter; then will CE* = CA*–CF2. For draw CE perpendicular to A B, and join FE. By Cor. 2. Th. II., CA2 : CE2:: CA2– CF2 : FG2, and the parameter FG is a third proportional to CA, CE; therefore CA2 : CE*:: CE2 : FG2, and as in the two ana- AH5 logies the first, second, and fourth terms are identical, the third > terms are equal; consequently CE2 = CA2- CF2. Fig. 412. Coroll. 1. Hence CF2 = CA2- CE2. Coroll. 2. The two semi-axes and the distance of the focus from the centre are the sides of a right-angled triangle CFE, and the distance FE from the focus to the extremity of the minor axis is equal to CA or CB, or to half the major axis. Coroil. 3. The minor axis CE is a mean proportional between the two segments of the axis on each side of the focus. For CE* = CA*–CF2=(CA + CF) x (CA-CF). 1069. THroREM. V. In an ellipsis, the sum of the lines drawn from the foci to any point in the curve is equal to the major aris. Let the points F, f(fig. 413.) be the two foci, and M a point in the curve; join FM and fM, then will AB=2CA = FM +f M. By Cor. 2. Th. II., CA2 : CE2:: CA2–CP2: PM2, But by Th. IV., CE2 = CA2–CF2; Therefore CA2 : CA2–CF2:: CA2–CP2: PM2; Fig. 415. And by taking the rectangle of the extremes and means, and dividing the equation by CA2, the result is— these is FM +f M=2CA. Coroll. 1. A line drawn from a focus to a point in the curve is called a radius vector, and the difference between either radius vector and half the major axis is equal to half the difference between the radius vectors. For, since Coroll. 2. Because c£r is a fourth proportional to CA, CF, CP; therefore CA : CF:: CP: CA-fM. the other radius vector. For, since FM + Mf=2CA; Coroll. 4. Hence is derived the common method of describing an ellipsis mechanically, by a thread or by points, thus: – Find the foci Ff (fig. 414.), and in the axis AB assume any point G; then with the radius AG from the point F as a hi E h centre describe two arcs H, H, one on each side of the axis; and / - - ~ with the same radius from the point f describe two other arcs h, A. h, one on each side of the major axis Again, with the distance G ic J/ GB from the point f describe two arcs, one on each side of the axis, S. intersecting the arcs HH in the points HH ; and with the same h radius from the point f describe two other arcs, one on each side of Fig. 414. the axis, intersecting the arcs described at h, h in the point h, h. In this manner we may find as many points as we please; and a sufficient number being found, the curve will be formed by tracing it through all the points so determined. 1070. THroREM VI. The square of half the major aris is to the square of half the minor aris as the difference of the squares of the distances of any two ordinates from the centre to the difference of the squares of the ordinates themselves. Let PM and HI (fig. 415.) be ordinates to the major axis AB; draw MN parallel to AB, meeting HI in the point N; then will PM = HN, and MN=PH, and the property to be demonstrated is thus expressed- Fig. 415. or by producing HI to meet the curve in the point K, and making CQ=CP, the property to be proved will be 1071. TheoREM VII. In the ellipsis, half the major aris is a mean proportional between the distance of the centre and an ordinate, and the distance between the centre and the intersection of a tangent to the verter of that ordinate. To the major axis draw the ordinates PM (fig. 416.) and HI, . and the minor axis CE. Draw MN perpendicular to HI. Through the two points I, M. draw MT, IT, meeting the major axis produced in T; then will CT : CA:: CA : CP. For, Fig. 416. By Cor. 1. Theor. VI., CE2 : CA:::(IH + HN).IN : (PC+CH)HP; By Cor. 2. Th. II., CE2 : CA2::PM2 : CA2–CP2; Therefore, by equality, PM2 : CA2–CP2::(IH + HN)IN : (PC + CH)H P. By similar triangles, INM, MPT, IN : NM or PH::PM ; PT or CT-CP. Therefore, taking the rectangles of the extremes and means of the two last equations, and throwing out the common factors, they will be converted to the equation But when HI and PM coincide, HI and HN will become equal to PM, and CII will become equal to CP; therefore, substituting in the equation 2CP for CP + PH, and 2PM for IH + HN, and throwing out the common factors and the common terms, we have CT. CP = CA2 Coroll. 1. Since CT is always a third proportional to CP and CA, if the points P, A, B remain fixed, the point T will be the same; and therefore the tangents which are drawn from the point M, which is the intersection of PQ and the curve, will meet in the point T in every ellipsis described on the same axis A B. Coroll. 2. When the outer ellipsis AQB, by enlarging, becomes a circle, draw QT perpendicular to CQ, and joining TM, then TM will be a tangent to the ellipsis at M. Coroll. 3. Hence, if it were required to draw a tangent from a given point T in the prolongation of the major axis to the ellipsis AEB, it will be found thus: – On AB describe the semicircle AQB. Draw a tangent TQ to the circle, and draw the ordinate PQ intersecting the curve AEB of the ellipsis in the point M, join TM; then TM is the tangent required. This method of drawing a tangent is extremely useful in practice. 1072. THEoREM VIII. Four perpendiculars to the major aris intercepted by it and a tangent will be proportionals when the first and last have one of their P ertremities in the vertices, the second in the point of contact, and the third in the centre. Let the four perpendiculars be AD, PM, CE, BF, of which AD and BF have their extremities in the vertices A and B, the second in the point of contact M, and the third in the centre C; T then will But by the similar triangles TAD, TPM, TCE, and TBF, the sides TA, TP, TC, and Coroll. 2. The triangles APM and CBF are similar; Therefore, by equality, AP : PM :: CB : B F. |