文 [3]对文 [1 ]、[2 ]作了改进 ,用配方法证明了“函数y=(ax +c) + bx +d(a ,b ,c,d∈R且ab≠ 0 )的图象是双曲线” ,由于配方技巧较高 ,一般学生理解起来仍然很费力 ,本文作进一步改进 ,给出一种更简明的解析说明方法 .引理　与两相交定直线的距离之积等于正常数的动点的轨迹 In [3], improvements were made to [1] and [2], and the function "y=(ax + c) + bx + d(a, b, c, d∈R and ab≠ 0) was proved by the matching method. The image is hyperbolic. Due to the high formulation skills, the average student still understands it is still very laborious. This article makes further improvements and gives a more concise way to explain. The product of the distance between the lemma and the two intersecting straight lines is equal to Track of moving points of normal numbers