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一、对称思想方法对称是在数学、物理等自然科学乃至现实生活中广泛存在的一种现象,利用对称方法解高考数学试题主要是利用这些问题中蕴涵的数学对称性(如变量之间的交换对称性、几何图形的对称性等)来解题.常见的结论有: (1)若对任意的x∈R,函数f(x)总有f(a-t)=f(t+b)成立,则f(x)关于直线x=((a-t)+(b+t))/2=(a+b)/2对称。 (2)椭圆(双曲线)x~2/a~2+y~2/b~2=1((x~2/a~2)-(y~2/b~2))=1的方程的轨迹不但关于x轴而且关于y轴对称,所以若(x_0,y_0)是椭圆(双曲线)上的点,(则±x_0,±y_0)都是椭圆(双曲线)上的点。例1 若曲线x~2-y~2=a~2和(x-1)~2+y~2=1有三个交点,试求实数a的值。解析:一般的参考书常会用数形结合思想方法
First, the symmetry thinking method Symmetry is a phenomenon that exists extensively in mathematics, physics and other natural sciences and even in real life. Using symmetry methods to solve college mathematics test questions is mainly to use the mathematical symmetries contained in these problems (such as the exchange between variables Symmetry, geometry symmetry, etc.) to solve the problem. Common conclusions are: (1) For any x∈R, the function f(x) always has f(at)=f(t+b) established, Then f(x) is symmetric about the straight line x=((at)+(b+t))/2=(a+b)/2. (2) The equation of elliptic (hyperbolic) x~2/a~2+y~2/b~2=1((x~2/a~2)-(y~2/b~2))=1 The trajectory is not only about the x-axis but also about the y-axis, so if (x_0, y_0) is a point on the ellipse (hyperbolic), (±x_0, ±y_0) are all points on the ellipse (hyperbola). Example 1 If the curves x~2-y~2=a~2 and (x-1)~2+y~2=1 have three intersection points, try to find the value of the real number a. Analytical: General reference books often use a combination of number and shape ideas