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高中课本《平面解析几何》(甲种本)P74例3的解法欠妥。题目是:已知圆的方程x~2+y~2=r~2,求经过圆上一点M(x_0,y_0)的切线方程。教科书上的解法可简述为:设所求切线的斜率为k,根据切线垂直于过切点的半径求出k_0,从而得到所求切线方程为xx_0+yy_0=r~2。笔者认为,这种求解过程中忽视了一个特殊的情况,即M(x_0,y_0)是圆上任意一点,既然是任意的,试问,若点M(x_0,y_0)在x轴上时,能设所求切线的斜率吗? 要对一般情况进行讨论,得出一般性结论,就必须对特殊情况加以讨论,这是在解数学题中不能忽视的一个环节。因此,对例3的正确解法应分两个步骤处理: 1 若点M(x_0,y_0)不在x轴和y轴上时,可按教科书上的方法求解。
The solution to the P74 case 3 in the high school textbook “Plane Analytic Geometry” (A type) is not proper. The title is: Known circle equation x~2+y~2=r~2, find the tangent equation passing through a point M(x_0,y_0) on the circle. The textbook solution can be briefly described as follows: Let the slope of the tangent line be k, calculate k_0 based on the tangent perpendicular to the radius of the overcut point, and obtain the tangent equation xx_0+yy_0=r~2. The author believes that a special case is ignored in this solution process. That is, M(x_0, y_0) is an arbitrary point on the circle. Since it is arbitrary, ask, if the point M (x_0, y_0) is on the x-axis, Do you need to determine the slope of the tangent? If you want to discuss the general situation and come to a general conclusion, you must discuss the special situation. This is a link that can not be ignored in solving mathematical problems. Therefore, the correct solution to Example 3 should be handled in two steps: 1 If the point M (x_0, y_0) is not on the x-axis and y-axis, it can be solved by the textbook method.