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定义:连结椭圆上任意两点的线段叫弦.过椭圆中心的弦叫直径.类似地可定义双曲线的直径.如图1,平行于直径CD的弦的中点的轨迹AB和直径CD叫互为共轭直径.类似地可定义双曲线的共轭直径. 定理1 已知AB、CD为椭圆x~2/a~2+y~2/b~2=1的一对共轭直径,其斜率分别为k_(AB)、K_(CD),那么K_(AB)·K_(CD)=-b~2/a~2. 略证:如图1,设平行弦EF簇的斜率为k(即K_(CD)),则平行弦EF簇的方程为 y=kx+t(t为参数).① 又椭圆方程为 x~2/a~2+y~2/b~2=1. ② ①代入②整理得 (a~2k~2+b~2)x~2+2a~2tkx+a~2(t~2-b~2)=0. ③ 由韦达定理,得x_1+x_2=-(2a~2tk/a~2k~2+b~2). 设M(x′,y′)是EF的中点,则 x′=1/2(x_1+x_2)=-(a~2tk/a~2k~2+b~2) ④ 点M在EF上,则y′=kx′+t. ⑤ 由④、⑤消去参数t得 y′=-b~2/a~2k x′. ∵k_(AB)=k_(OM)=-(b~2/a~2k). ∴k_(AB)·k_(CD)=-(b~2/a~2k)·k=-(b~2/a~2). 推论1 AB是椭圆x~2/a~2+y~2/b~2=1的任意一条弦,P为AB的中点,O为椭圆的中心,则 K_(AB)·K_(OP)=-(b~2/a~2).
Definition: The line segment connecting any two points on the ellipse is called a string. The diameter of the line that passes through the center of the ellipse is called the diameter. Similarly, the diameter of the hyperbola can be defined. As shown in Fig. 1, the track AB and the diameter CD of the midpoint of the string parallel to the diameter CD are called. The conjugate diameters are similar to each other. Similarly, the conjugate diameter of the hyperbola can be defined. Theorem 1 It is known that AB and CD are a pair of conjugate diameters of elliptic x~2/a~2+y~2/b~2=1. The slopes are k_(AB) and K_(CD), then K_(AB)·K_(CD)=−b~2/a~2. Roughly: As shown in Figure 1, set the slope of the parallel chord EF cluster to k (ie K_(CD)), then the equation of the parallel string EF cluster is y=kx+t (t is the parameter). 1 The elliptic equation is x~2/a~2+y~2/b~2=1. 2 1 into the 2 finishing (a~2k~2+b~2)x~2+2a~2tkx+a~2(t~2-b~2)=0. 3 from the Weida theorem, get x_1+x_2 =-(2a~2tk/a~2k~2+b~2). Let M(x’,y’) be the midpoint of EF, then x’=1/2(x_1+x_2)=-(a~ 2tk/a~2k~2+b~2) 4 points M is on EF, then y’=kx’+t. 5 4 and 5 eliminate parameter t to get y’=-b~2/a~2k x’ ∵k_(AB)=k_(OM)=-(b~2/a~2k). ∴k_(AB)·k_(CD)=-(b~2/a~2k)·k=-(b ~2/a~2). Corollary 1 AB is any chord of the ellipse x~2/a~2+y~2/b~2=1, P is the midpoint of AB, and O is the center of the ellipse, then K_ (AB)·K_(OP)=-(b~2/a~ 2).