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我们知道:过两点A(x_1,y_1)和B(x_2,y_2)的直线与直线l:Ax+By+C=0交于点P。则P点分AB所成的比λ=-Ax_1+By_1+C/Ax_2+By_2+C.(Ⅰ).上式结构很有规律,因而便于记忆和应用。它的右端只是把A、B两点的坐标分别代入直线l的方程的左端求值,并取两者商的相反数。因此我们不妨把这个式子称为直线分有向线段比的公式。问题是:若直线l换成二次曲线C,即有向线段所在直线与一条二次曲线相交,交点分有向线段的比是否也有类似的式子呢?答案是肯定的,下面我们给出这个式子,并加以应用。
We know that the two lines A (x_1, y_1) and B (x_2, y_2) intersect with the straight line l:Ax+By+C=0 at point P. The ratio of P points to AB is λ=-Ax_1+By_1+C/Ax_2+By_2+C. (I). The above structure is very regular, so it is easy to remember and apply. At the right end of the equation, only the coordinates of the two points A and B are respectively substituted into the left end of the equation of the straight line l, and the opposite of the two quotients is taken. Therefore, we may wish to call this formula a formula for the ratio of straight lines to directed lines. The question is: if the straight line l is replaced by a quadratic curve C, that is, the line where the line segment is directed intersects with a quadratic curve, does the ratio of the intersecting points to the line segment also have similar formulas? The answer is yes. This formula is applied.