THE GEOMETRY OF CONICS. INTRODUCTION. THE name CONIC is applied to a family of curves to which the circle belongs. It includes three varieties, the Ellipse, the Parabola, and the Hyperbola. THE ELLIPSE is the simplest in form. Its relation to the circle is shewn by the following construction, which will be found to lead to important results. From points in a circle let fall perpendiculars pN, dR,...to a fixed diameter AA'. If these perpendiculars be cut in a constant ratio the points of section will lie on an ellipse. [73. By making the constant ratio of PN to pN very small we may flatten the ellipse indefinitely. And by making the ratio approximate to a ratio of equality we may make the ellipse as nearly circular as we please. A construction much simpler in practice results from using two circles, thus: Let AA', BB' be diameters at right angles of two concentric circles. If from the points in which a common radius cuts the circles parallels be drawn to BB', AA' respectively, their intersection will describe an ellipse. [Ex. 130. The following construction shews the relation of the ellipse to two points called the foci*. Let a loop of string, The planets describe approximately ellipses having the sun in one focus. For this reason the first letter of Sol is used, as by Newton, to denote a focus. supposed inelastic, be passed over two pins at points S, S (fig. p. 47), and let it be stretched into the form of a triangle, always in the same plane, by the point of a pencil or sharp instrument at P. Then as P moves in such a way as to keep the string stretched it describes an ellipse. It is evident that if S, S' be made to coincide the ellipse becomes circular, and if the distance SS' be made as great as possible, i. e. equal to half the length of the loop, the ellipse becomes flattened indefinitely. Example 98 gives another construction for the ellipse. THE PARABOLA is the curve which would be described by a particle moving in a vacuum under the influence of gravity. A practical construction is given in Example 11. THE HYPERBOLA is most simply drawn by the analogous construction of Example 213. THE CONE. The three curves considered above were originally treated as plane sections of a Cone. Hence their old name Conic Sections. The cone and its sections may be shewn by means of a wooden model. An ellipse may also be cut from a cylinder or roller of circular transverse section. If the roller be cut obliquely, the section, supposed plane, will always be an ellipse. We may shew the sections optically by casting the shadow of a sphere or of a circular disc from a point of light upon a plane surface*. If the point of light be vertically under a point in the rim of the disc, the shadow thrown upon a vertical wall will be parabolic. These constructions and illustrations will suffice to give a practical acquaintance at the outset with the forms of Conics. We shall not, however, in our fundamental definition make use of any of the constructions already given. * They may also be shewn very roughly by means of a light held at the small end of a conical lamp shade. LEMMAS. A. To prove geometrically that 2 2 (a + b)2 - (a ~ b)2 = 4ab. If four rectangles of sides a, b be fitted symmetrically about the square on a ~b, the whole figure thus made up will be the square on a+b. or Therefore (a+b)2 = (a ~ b)2 + 4ab, (a + b)3 — (a~b)2 = 4ab. The same is proved in Euc. II. 8, but by an unsymmetrical construction which shews only a gnomon of equal area instead of the four rectangles. B. From the ends of a straight line QQ and from its middle point v (fig. p. 6) let parallels QM, Q'M', vO be drawn to meet any other straight line. Draw parallels from Q', Q to M'M, and let them meet vO in Z', Z. If the figure be drawn as on p. 66, where O bisects Qq, it may be shewn in like manner that QM — qm = 2OL. If the parallels drawn as above be called the ordinates of the points from which they are drawn, we may enunciate the Lemma as follows: The ordinate of the centre of a straight line is equal to half the sum or difference of the ordinates of its ends. As a particular case of the above, if S be a point in the straight line itself, then For, SQ ± SQ' = 2Sv. SQ — Sv = & Q Q' = Sv + SQ′ ; and similarly when S cuts QQ' externally. [Fig. p. 6. C. In the figure of Art. 78, let PN be drawn perpen- dicular to SS". and 2 SP2 = CS2+ CP + 2CS. CN, S'P2=CS"2+ CP2-2 CS'. CN. By addition, if C bisects SS', SP2 + S'P2 = 2 CP2 +2CS2. D. In Art. 96, let a parallel from 0 to RS meet SP in M. Then SM OR SP: PR. = And because the bisector of the angle PSQ is perpendicular to OM, and is nearer to SO than to SM; therefore SO is less than SM. Therefore SO: OR < SP : PR. DEFINITION. The term Equivalent will be used in this work to denote equal and similar or equal in all respects. CHAPTER I. CHORD-PROPERTIES OF THE PARABOLA. DEF. A parabola is the curve described in a plane by a point which moves in such a way that its distance from a certain fixed point, called the Focus, is always equal to its perpendicular distance from a certain fixed straight line, called the Directrix. 1. With any straight line MX as directrix, and any point S exterior to it as focus, a parabola may be described. M M XAS From any point M on the directrix draw a straight line at right angles to the directrix. On the straight line so drawn one point P belonging to the parabola can be found; for if the angle MSP be made equal to the known angle MSX, or SMP, then SP=PM, or P is a point on the curve. In this way any number of points P, P' belonging to the parabola may be found; and they all lie on the same side of the directrix with the focus. The straight line through the focus at right angles to the directrix is called the Axis. The point in which the axis meets the parabola is called the Vertex. Let the axis meet the directrix in X. The vertex A is the centre of the straight line SX, since from the definition the distance of A from S must be equal to its perpendicular distance AX from the directrix. |