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1问题背景在圆锥曲线问题中,“点差法”即代点相减法,对于解决中点弦方程、弦中点轨迹方程以及对称问题等方面都非常有效,堪称利器~([1,2]).然而,“点差法”有时却会失效,导致错误的答案~([3]).以笔者所在学校的高二数学月考题为例:例1已知直线l经过定点(0,1),被双曲线x~2-y~2/4=1所截得的弦的中点轨迹方程是.大部分学生由“点差法”可求出轨迹方程,结果是4x~2-y~2+y=0,但正确答案是4x~2-y~2+y=0,y∈(-∞,
1 Problem Background In the conic curve problem, “Spread Method ” that is, on behalf of the point subtraction method, for solving the mid-point chord equation, chord midpoint trajectory equation and symmetry problems are very effective, can be called sharp weapon ~ ([1, 2]). However, “spread” sometimes fails, leading to the wrong answer ~ ([3]). In my author’s school mathematics month exam questions as an example: , 1), the midpoint locus equation of the string which is intercepted by the hyperbola x ~ 2-y ~ 2/4 = 1 is that most of the students can find the locus equation from "point difference method 2-y ~ 2 + y = 0, but the correct answer is 4x ~ 2-y ~ 2 + y = 0 and y∈ (-∞,