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一阶导数与二次曲线弦中点间存在着一种内在联系,这种联系为解决二次曲线中点弦一类问题开辟了一条较为简捷的路径.本文就以定理形式揭示这种联系并列举应用. 定理:椭圆x~2/a~2+y~2/b~2=1的以斜率为k的一组平行弦中点轨迹方程是x~2/a~2+yy_x~’/b~2=0(※)(|x|≤a,|y|≤b)其中y_x~’就是平行弦的斜率k,它等于直线(※)与椭圆交点处切线的斜率. 证明:设点P(x_0,y_0)是以k为斜率的弦P_1P_2的中点,点P_1(x_1,y_1),P_2(x_2,y_2)
There is an intrinsic link between the first derivative and the midpoint of the quadratic curve. This connection opens up a relatively simple path for solving the midpoint chord problem of the quadratic curve. This article reveals this connection in the form of a theorem. List the applications. Theorem: The midpoint trajectory equation of a set of parallel chords with a slope of k is an x=2/a~2+yy_x~/ b~2=0(*)(|x|≤a,|y|≤b) where y_x~’ is the slope of the parallel string, k, which is equal to the slope of the tangent line at the intersection of the straight line (*) and the ellipse. Proof: set point P(x_0, y_0) is the midpoint of the string P_1P_2 with the slope of k, the point P_1(x_1,y_1), P_2(x_2,y_2)