CONIC SECTIONS

In Geometry, lines formed by the intersections of a plane with the surface of a cone, which a ume different forms, and acquire different properties, according to the different positions of the plane m respect o"f the axis of the cone. There are five species: — 1. If the cond be cut by a plane pa ing through its vertex, the common intersection o« the conical superficies and the plane will be two straight lines meeting in the vertex. 2. If the intersecting plane be parallel to the base, or, in the case of the oblique or scalene cone, if it be so situated as to cut off from the vertex a cone similar to the original cone, the section will be a circle. 3. If the intersecting plane be parallel to a plane which touches the cone, the section will be a parabola. 4. If the intersecting plane pa through both sides of the cone, and is neither parallel to the base nor to the plane of a subcontrary section, the section will be an ellipse. 5. If the intersecting plane have such a position that, when produced, it meets the opposite cone, the section is a hv'perbola. These five are the only lines which can be formed by the common intersection of a plane and the surface of a cone, and they all equally arise from that intersection; but as the straight line and the circle form the peculiar subject of elementary geometry, their properties are usually treated apart; and the three last, namely, the parabola, the ellipse, and the hyperbola, considered as the curves especially designated by the term Conic Sections. Some of the principal and distinctive properties of the curves are easily deduced from this mode of generation. Let V A C B be the cone, and C D E the section made by a plane parallel to the plane which touches the cone in the line VA; then, by the definition, C E D is a parabola. Now let P Q be the intersection of the plane which touches the cone in V A with the plane of the base A C B; then P Q being a tangent to the circle A C B D, is perpendicular to A B, the diameter of that circle, and consequently C D, which is parallel to P Q, is also perpendicular to A B j therefore C F is equal to F D. In F E take any point G, through which let there pa a plane H L K, parallel to the base A C ^, intersecting the plane C E D in the straight line L G M; then L M will be parallel to C D, and perpendicular to H K, and L G equal to G M. We have therefore, from the property of the circle, C F2 = A F • F B, and L G^ = H Gf • G K; therefore since by reason of the parallels V A and E F the line A F is equal H G, C Y^: L G^::FB: GK. ButFB:GK::EF:EG; therefore C F2: L G2:: E F: E G; consequentlv, since C F and E F are constant quantities, the ratio of L G^ to E.G (or of L M2 to E G) is constant; whence we infer that in the parabola the square of any ordinate L G is equal to the rectangle of the corresponding absci E G into a constant quantity. From this all the other properties of the curve may be deduced. It is in fact the common equation of the parabola. See Parabola. [s. 284]