Page If TP, TQ be tangents to opposite branches, then STP = HTQ 104 The foot of the focal perpendicular lies on the circle upon the axis GEOMETRICAL CONICS. DEFINITIONS. A CONIC is the curve traced out by a point which moves in such a way that its distance from a fixed point, called the Focus, bears always the same ratio to its perpendicular distance from a fixed straight line, called the Directrix. M Note. Let S be the focus, MX the directrix, P and P' any two points on a given conic. Draw SX, PM, P'M' perpendicular to the directrix, and through P, P' draw any two parallel straight lines meeting the directrix in R, R'. Then by similar triangles PMR, P'M'R', M R R P' Χ S [Euc. v., 22. Now P' may be any point on the conic, but, whatever be the position of P', the ratio SP' : P'R' is always equal to the ratio SP: PR. Hence a conic might have been defined as the curve traced out by a point which moves in such a way that its distance from the focus bears always the same ratio to its distance from the directrix, measured parallel to any fixed straight line which meets the directrix. The B ordinary definition results from supposing this straight line perpendicular to the directrix. The Eccentricity of a conic is the ratio which the distance from the focus, of any point on the curve, bears to its perpendicular distance from the directrix. A conic is called a Parabola, Ellipse, or Hyperbola, according as its eccentricity is equal to, less or greater than unity. The Axis is a straight line drawn from the focus perpendicular to the directrix, and the point in which it intersects the conic is called the Vertex. The straight line joining any two points on a conic is said to be a Chord of the conic. The Latus Rectum is the chord drawn through the focus at right angles to the axis. Let P, Q be adjacent points on a conic, as in Prop. I., p. 6, and let move along the curve towards P, whilst P remains stationary. Then the chord PQ, in its limiting position, when coincides with P, becomes the Tangent at P. The Normal at any point of the curve is the straight line drawn through that point at right angles to the tangent. The perpendicular upon the axis from any point of the curve is said to be the Ordinate of the point. The portion of the axis intercepted between the tangent and ordinate at any point on the curve is called the Subtangent. The portion of the axis intercepted between the normal and ordinate at any point on the curve is called the Subnormal. CHAPTER I. TRACING THE CURVE. 1. When the focus, directrix, and eccentricity of a conic are given, any number of points on the curve may be determined. For, let S be the focus, MM' the directrix. Draw SX meeting the directrix at right angles in X, and in SX take a point A such that the ratio of SA to AX may be equal to the eccentricity. Then A is the vertex of the curve. In AS, or AS produced, take any point N, and with centre S, radius SP, such that SP: NX= SA : AX describe a circle cutting in P, P', the straight line M M A S N drawn through N parallel to the directrix. Let PM, P'M' be the perpendiculars from P, P' on the directrix. Then PM is equal to NX. Thus any number of points on the curve may be determined corresponding to the various positions of N. |