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2016年全国高中数学联赛一试的最后一题为:如图1所示,在平面直角坐标系xOy中,F是x轴正半轴上的一个动点。以F为焦点、O为顶点作抛物线C。设P是第一象限内C上的一点,Q是x轴负半轴上的一点,使得PQ为抛物线C的切线,且|PQ|=2。圆C_1,C_2均与直线OP相切于点P,且均与x轴相切。求点F的坐标,使圆C_1与C_2的面积之和取到最小值。这是近几年来全国高中数学联赛解析几何试题中较难的一道题,难度系数不足0.2。本题难点分
In the 2016 National High School Maths League, the final question was: As shown in Figure 1, in a Cartesian coordinate system xOy, F is a moving point on the positive x-axis. F as the focus, O as the vertex for the parabola C. Let P be a point on C in the first quadrant and Q be a point on the negative half-axis of the x-axis such that PQ is the tangent to parabola C and | PQ | = 2. The circles C_1, C_2 are tangent to the line OP at point P, and are tangent to the x axis. Find the coordinates of point F so that the sum of the areas of circles C_1 and C_2 takes the minimum value. This is in recent years the National High School Mathematical Olympiad analytical geometry problem more difficult problem, the difficulty coefficient of less than 0.2. Difficult points of this problem