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椭圆以某定点为中点的弦并非一定存在,那么,中点弦存在的充要条件是什么?有何应用,本文作下列探讨: 一中点弦方程的一种求法。设椭圆b~2x~2+a~2y~2-a~2b~2=0,(a>0,b>0)…(1) 及定点P_0(x_0,y_0),若以P_0为中点的弦存在,且两端点分别为A(x_1,y_1),B(x_2,y_2) 则:b~2x_1~2+a~2y_1~2-a~2b~2=0 b~2x_2~2+a~2y_2~2-a~2b~2=0 两式相减整理得: (y_1-y_2)/(x_1-x_2)=(x_1+x_2)/(y_1+y_2)·b~2/a~2 =-b~2/a~2·x_0/y_0 (x_1≠x_2) 即k=-(b~2x_0)/(a~2y_0),代入点斜式得中点弦方程:a~2y_0y+b~2x_0x=a~2y_0~2+b~2x_0~2……(2) 如果x_1=x_2,那么y_0=0,中点弦方程为x=x_0仍包含在(2)中。
The midpoint of the ellipse does not necessarily exist as a midpoint point. What is the necessary and sufficient condition for the existence of the midpoint chord? What are the applications? This article discusses the following: A method for finding a midpoint string equation. Let ellipsoid b~2x~2+a~2y~2-a~2b~2=0, (a>0, b>0)...(1) and fixed point P_0(x_0,y_0), if P_0 is the midpoint The string exists, and the two ends are A(x_1,y_1), B(x_2,y_2) then: b~2x_1~2+a~2y_1~2-a~2b~2=0 b~2x_2~2+a ~2y_2~2-a~2b~2=0 Two types of subtraction are: (y_1-y_2)/(x_1-x_2)=(x_1+x_2)/(y_1+y_2)·b~2/a~2 =-b~2/a~2*x_0/y_0 (x_1≠x_2) That is, k=-(b~2x_0)/(a~2y_0). Substitution point is skewed to midpoint. String equation: a~2y_0y+b~ 2x_0x=a~2y_0~2+b~2x_0~2 (2) If x_1=x_2, then y_0=0 and the midpoint string equation x=x_0 is still included in (2).