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抛物线y~2=2px的焦点弦为AB,则y_Ay_B=-p~2,这是抛物线焦点弦的一条常用性质.对一般的弦而言,也有类似的性质,这里,我们给出一组充要条件,揭示弦的性质. 若AB为抛物线y~2=2px的弦,其中A(x_1,y_1)、B(x_2,y_2).则有: ∠AOB为直角x_1x_2+y_1y_2=0 y_1y_2+Ap~2=0; ∠AOB为锐角x_1x_2+y_1y_2>0 y_1y_2(y_1y_2+4p~2)>0; ∠AOB为钝角x_1x_2+y_y_2<0 y_1y_2(y_1y_2+4p~2)<0. 证明:cos∠AOB=|AO|~2+|BO|~2-|AB|~2/2|AO|·|BO|=2(x_1x_2+y_1y_2)/2|AO|·|BO|,故∠AOB为直角cos∠AOB=0x_1x_2+y_1y_2=0; ∠AOB为锐角cos∠AOB>0 x_1x_2+y_1y_2>0; ∠AOB为钝角cos∠AOB<0 x_1x_2+y_1y_2<0. 又A、B在抛物线上,故y_1~2=2px_1,y_2~2=2px_2,从而(y_1y_2)~2=4p~2x_1x_2,故x_1x_2+y_1y_2=1/4p~2·y_1y_2(y_1y_2+4p~2). 从而 x_1x_2+y_1y_2=0 y_1y_2+4p~2=0(显然y_1y_2≠0), x_1x_2+y_1y_2>0 y_1y_2(y_1y_2+4p~2)>0, x_1x_2+y_1y_2<0 y_1y_2(y_1y_2+4p~2)<0,得证. 应用这组充要条件,可方便地解决与抛物线弦相关的一类问题.
The focus of the parabola y~2=2px is AB, then y_Ay_B=-p~2, which is a common property of the parabola focus chord. For the general chord, there are similar properties. Here, we give a group of charge To qualify, reveal the properties of the string. If AB is a parabola with y~2=2px, where A(x_1,y_1), B(x_2,y_2). Then: ∠AOB is a right angle x_1x_2+y_1y_2=0 y_1y_2+Ap ~2=0; ∠AOB is the acute angle x_1x_2+y_1y_2>0 y_1y_2(y_1y_2+4p~2)>0; ∠AOB is the obtuse angle x_1x_2+y_y_2<0 y_1y_2(y_1y_2+4p~2)<0. Proof: cos∠AOB =|AO|~2+|BO|~2-|AB|~2/2|AO|·|BO|=2(x_1x_2+y_1y_2)/2|AO|·|BO|, so ∠AOB is a right angle cos ∠AOB=0x_1x_2+y_1y_2=0; ∠AOB is the acute angle cos∠AOB>0 x_1x_2+y_1y_2>0; ∠AOB is the obtuse angle COS∠AOB<0 x_1x_2+y_1y_2<0. Also, A and B are on the parabola, so y_1~ 2=2px_1, y_2~2=2px_2, thus (y_1y_2)~2=4p~2x_1x_2, so x_1x_2+y_1y_2=1/4p~2·y_1y_2(y_1y_2+4p~2). Thus x_1x_2+y_1y_2=0 y_1y_2+4p ~2=0 (obviously y_1y_2≠0), x_1x_2+y_1y_2>0 y_1y_2(y_1y_2+4p~2)> 0, x_1x_2+y_1y_2<0 y_1y_2(y_1y_2+4p~2)<0, obtained. Apply this charge For conditions, a class of problems related to parabolic strings can be easily solved.