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中点弦问题是解析几何中的重点、热点问题.解圆锥曲线的中点弦问题,很多学生习惯于用所谓“点差法”:首先设出弦的两端点坐标,然后代人圆锥曲线方程相减,得到弦中点的坐标与所在直线的斜率的关系,从而求出直线方程.但是,有时候符合条件的直线是不存在的,这时使用“点差法”便会走入“误区”.下面问题中便有学生经常掉入“陷阱”.题目:已知双曲线 x~2-y~2/2-1,问是否存在直线 l,使 M(1,1)为直线 l 被双曲线所截弦 AB 的中点.若存在,求出直线 l 的方程;若不存在请说明理由.错误解法1:(点差法)设直线与双曲线两交点 A、B 的坐标分别为(x_1,y_1),(x_2,y_2),M 点的坐标为(x_M,y_M).由题设可知直
Mid-point string problem is the focus of analytic geometry, hot issues. Solution of the mid-point chord conic problem, many students are accustomed to using the so-called “spread method”: first set the coordinates of the two ends of the string, and then replace the conic The equation is subtracted to get the relationship between the coordinates of the middle point of the string and the slope of the line where it is located, so as to find the straight line equation. However, sometimes the line that does not meet the requirement does not exist. “Mistakes ”. The following questions will have students often fall into the “Trap ” Title: known hyperbola x ~ 2-y ~ 2 / 2-1, asked whether there is a straight line l, so that M 1) is the midpoint of the line l being truncated by the hyperbola AB, if there is, find the equation of the line l, if not, please explain the reason Error Solution 1: (point spread method) Set the intersection of the line and the hyperbola A, The coordinates of B are (x_1, y_1), (x_2, y_2), and the coordinates of point M are (x_M, y_M)