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求圆的方程的基本方法是待定系数法.若已知条件与圆心、半径有关,可设圆的方程为标准式,建立关于a、b、r的方程组,解出待定系数a、b、r即可;若已知条件涉及到圆过几个点,则常用圆的一般方程,建立关于D、E、F的方程组,解出待定系数D、E、F而获解;若所求的圆过两已知圆C1、C2的交点(或一直线与一圆的交点),一般用共轴圆系C1+λC2=0,建立方程f(λ)=0,解出λ即可得到所求圆方程.但如何构建关于待定系数a、b、r或D、E、F的方程组和关于λ的方程,则是解题成败的关键.本文仅就构建这类方程(组)的几种常见技巧例示如下.
The basic method for finding the circular equation is the undetermined coefficient method. If the known conditions are related to the center of the circle and the radius, the equation of the circle can be set as the standard equation. The equations for a, b, and r are established and the coefficients to be determined a and b are solved. r can; if the known conditions involve rounding a few points, the common equation of the circle is commonly used to establish a system of equations for D, E, and F, and the undetermined coefficients D, E, F are solved to obtain solutions; The circle intersects the intersection of two known circles C1 and C2 (or the intersection of a straight line and a circle). Generally, the coaxial circle C1+λC2=0, the equation f(λ)=0 is established, and the solution of λ is obtained. The equation of the circle is required. However, how to construct a system of equations about the undetermined coefficients a, b, r or D, E, F and an equation about λ is the key to the success or failure of the problem. This paper only constructs the equations (groups) of this type. Several common techniques are illustrated below.