In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In this paper "Some new identities on the conic sections", we construct some new identities and proposed the concept of the power of a point with respect to a conic. | Journal of Science and Arts Year 14 No. 3 28 pp. 199-210 2014 ORIGINAL PAPER SOME NEW IDENTITIES ON THE CONIC SECTIONS DAM VAN NHI 1 TRAN TRUNG TINH1 PHAM MINH PHUONG 2 _ Manuscript received Accepted paper Published online . Abstract. In mathematics a conic section or just conic is a curve obtained as the intersection of a cone more precisely a right circular conical surface with a plane. In this paper we construct some new identities and proposed the concept of the power of a point with respect to a conic. Keywords Conic section identity power line power of a point. 2010 Mathematics Subject Classification 26D05 26D15 51M16. 1. THE ECCENTRICITY OF CONIC SECTION Definition . A conic section or conic is a curve in which a plane not passing through the cone s vertex intersects a cone. Conics possess a number of properties one of them consisting in the following result. Proposition . 2 Each conic section except for a circle is a plane locus of points the ratio of whose distances from a fixed point F and a fixed line d is constant. The point F is called the focus of conic the line d its directrix. Proof Let be the curve in which the plane P intersects a cone. We inscribe a sphere in the cone which touches the plane P at the point F . Let ω be the plane containing the circle along which the sphere touches the cone. We take an arbitrary point M and draw through it a generator of the cone and denote by B the point of its intersection with the plane ω . We then drop a perpendicular from M to the line d of intrsection of the planes P and ω example MA d . We obtain FM BM because they are the tangents to the sphere drawn from one point. Further if we denote by h the h h distance of M from the plane ω then AM BM where α is the angle sin α sin β between the planes ω and P and β is the angle between the generator of the cone and AM AM sin β the ω . Hence it follows that . FM BM sin α AM sin β Thus the ratio λ does not depend on the point M .
