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定理设二次曲线方程为F(x,y)=Ax~2+2Bxy+Cy~2+2Dx+2Ey +F=0。(1)过平面上任意一定点M(x_0,y_0)(除去曲线的中心)作动直线,与曲线(1)交于P_1、P_2两点,则弦P_1P_2的中点轨迹方程是Φ(x-x_0,y-y_0)÷F_1(x_0,y_0)(x-x_0) ÷F_2(x_0,y_0)(y-y_0)=0(2)并且曲线(1)与曲线(2)同族。其中Φ(x,y)=Ax~2+2Bxy+Cy~2 F_1(x,y)=Ax+By+D F_2(x,y)=Bx+Cy+E 证明:设过定点M(x_0,y_0)的动直线为
Theorem Let the quadratic curve equation be F(x,y)=Ax~2+2Bxy+Cy~2+2Dx+2Ey+F=0. (1) A certain point M(x_0, y_0) (excluding the center of the curve) on the passing plane acts as a moving straight line and intersects the curve (1) at two points P_1 and P_2. The midpoint trajectory equation of the string P_1P_2 is Φ(x). -x_0, y-y_0) ÷ F_1 (x_0, y_0) (x-x_0) ÷ F_2 (x_0, y_0) (y-y_0)=0 (2) and curve (1) is the same family as curve (2). Where Φ(x,y)=Ax~2+2Bxy+Cy~2 F_1(x,y)=Ax+By+D F_2(x,y)=Bx+Cy+E Proof: Set the over-point M (x_0, The straight line for y_0) is