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韦达定理是代数中的一个重要定理,它在解析几何中也有广泛的应用。在解某些解析几何题时,如果注意运用韦达定理,有时能使运算简便。如以下几例。 一、利用x_1+x_2=-b/a 例1.点P(2,2)是椭圆x~2+8y~2+4x-24y+6=0的一条弦的中点,求这条弦所在的直线方程。 解:设所求的直线方程为y-2=k(x-2),它与椭圆的方程x~2+8y~2+4x-24y+6=0组成方程组,消去y得:(1+8k~2)x~2-(32k~2-8k-4)x+32k~2-16k-10=0,设它的两个根是x_1和x_2,则有x_1+x_2=4,根据韦达定理有
The Weida Theorem is an important theorem in algebra, and it is also widely used in analytical geometry. In solving some analytical geometry problems, if you pay attention to the use of Weida theorem, it can sometimes make the operation simple. As the following examples. 1. Using x_1+x_2=-b/a Example 1. Point P(2,2) is the midpoint of a string with the elliptic x~2+8y~2+4x-24y+6=0, and finds where this string is located. The equation of the line. Solution: Let the desired linear equation be y-2=k(x-2), and the equation of the ellipse x~2+8y~2+4x-24y+6=0 constitute the equation set. Eliminate y: (1 +8k~2)x~2-(32k~2-8k-4)x+32k~2-16k-10=0, set its two roots to be x_1 and x_2, and then have x_1+x_2=4, according to Weida Theorem