2005年北京春考理科第18题是一道解析几何综合题,我们做一些分析,会有所启示。试题如图1,O为坐标原点,直线l在x轴和y轴上的截距分别是a和b(a＞0,b≠0),且交抛物线y~2=2px(p＞0)于M(x_1,y_1),N(x_2,y_2)两点。 (Ⅰ)写出直线l的截距式方程； (Ⅱ)证明：1/y_1+1/y_2=1/b； (Ⅲ)当a=2p时,求∠MON的大小。试题叙述简洁明快,形式新颖。试题第(Ⅱ)问最初来源于对下面习题的改造：习题如图2,设抛物线y=ax~2与直线y=bx+c有两个交点,其横坐标分别为x_1,x_2,且a≠0,b≠0,b~2+4ac＞0,x_3是直线y=0与y=bx+c The 18th question of the Beijing Kaike Science Section in 2005 is a comprehensive analytical geometry problem. We will do some analysis and we will inspire it. The test questions are shown in Fig. 1. O is the origin of the coordinates. The intercepts of the straight line l on the x-axis and y-axis are a and b (a>0, b≠0), and the parabola y~2=2px (p>0). For M(x_1,y_1),N(x_2,y_2) two points. (I) Write the intercept equation of the straight line l; (II) Proof: 1/y_1+1/y_2=1/b; (III) When a=2p, find the size of ∠MON. The test descriptions are simple and neat, and the forms are novel. The first question (II) of the test question originally originated from the modification of the following exercises: Exercises are shown in Figure 2. Let the parabola y=ax~2 have two intersections with the straight line y=bx+c, and the abscissas are x_1, x_2, respectively, and a ≠0, b≠0, b~2+4ac>0, x_3 is a straight line y=0 and y=bx+c