ss_9_1 - Ax 2 Bxy Cy 2 Dx Ey F = 0 is a conic in the...

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Section 9.1 Conics Conics and conic sections are the names ordinarily applied to the class of plots that include parabolas, ellipses, and hyperbolas. Conics can be obtained in the following equivalent ways: (1) The set of points obtained by intersecting a plane with a right circular cone. (2) The set of all points P = P ( x, y ) in the xy - plane satisfying the equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 for some choice of constants A , B , C , D , E , and F . (3) The set of all points P = P ( x, y ) in the xy -plane such that d ( P, P o ) = kd ( P, L o ) where d ( P, P o ) denotes the distance from P to the fixed point P o , d ( P, L o ) denotes the distance from P to the fixed line L o , and k is a fixed constant, and P o does not lie on L o . Note: Just as the graph of the linear equation Ax + By + C = 0 is a line in the xy - plane, the graph of the quadratic equation
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Unformatted text preview: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 is a conic in the xy-plane. Degenerate Forms of Conics Degenerate forms can be obtained from the above, and degenerate forms can be graphs of no points, one point, a line, a pair of intersecting lines, or a circle. Identifying Conics The conic given by the equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 can be identified by the value of its discriminant B 2-4 AC as follows: B 2-4 AC = 0 Parabola or degenerate form B 2-4 AC < 0 Ellipse or degenerate form B 2-4 AC > 0 Hyperbola or degenerate form The following illustrates how the various conics are obtained from the intersections of planes with a cone. 1...
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