论文部分内容阅读
二次曲线系Ax~2+Bxy+cy~2+Dx+Ey+λ=0(其中A、B、C不全为零,λ是参变数,下同)有一些重要性质,值得研究。定理1 二次曲线系Ax~2+Bxy+Cy~2+Dx+Ey+λ=0……(1)中的非退化二次曲线是下列三种情形之 1°当B~2-4AC<0时,是一簇同轴(指对称轴)位似椭圆; 2°当B~2-4AC>0时,是一簇共渐近线双曲线; 3°当B~2-4AC=0时,是一簇同轴(指对称轴)同p(焦点到准线的距离)抛物线。证明1°当B~2-4AC<0时,曲线系(1)中的所有非退化二次曲线均是椭圆,它们经过适当的坐标变换,总可以化成最简椭圆方程;又因为经过坐标变换得到的新方程的二次项系数和一次项系数只与原方程的二次
The quadratic curve system Ax~2+Bxy+cy~2+Dx+Ey+[lambda]=0 (where A, B, and C are not all zero, and λ is a parameter, the same below) has some important properties and is worthy of study. Theorem 1 The non-degenerate quadratic curve of the quadratic curve system Ax~2+Bxy+Cy~2+Dx+Ey+[lambda]=0 (1) is 1 degree of the following three cases when B~2-4 AC< At 0, it is a cluster of coaxial (symmetry axis) like ellipses; 2° when B~2-4AC > 0, it is a cluster of asymptotic hyperbolic curves; 3° when B~2-4AC=0 , is a cluster of coaxial (referring to the axis of symmetry) and p (the distance from the focus to the guideline) parabola. Prove that 1 ° when B ~ 2-4AC <0, all the non-degenerate quadratic curve curve (1) are elliptic, they undergo proper coordinate transformation, can always be converted into the most simple elliptic equation; but also because of the coordinate transformation The quadratic coefficient and the first-order coefficient of the new equation are only quadratic to the original equation.