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焦点三角形是指以椭圆(或双曲线)的焦距F1F2为底边,顶点P在椭圆(或双曲线)上的三角形.熟练掌握焦点三角形的性质,对培养创新能力和解题能力具有重要意义.例题双曲线x29-y216=1的焦点为F1、F2,点P在双曲线上,若PF1⊥PF2,则点P到x轴的距离为.分析设P(x0,y0),则|y0|就是点P到x轴的距离,故只需求出点P的纵坐标即可.解法1(辅助圆法)构造以焦点F1、F2为直径的辅助圆.由圆的知识可知,若点P在圆上,则F1PF2是直角三角形;若点P在圆内,则F1PF2是钝角三角形;若点P在圆外,则F1PF2是锐角三角形.
The focus triangle refers to the triangle with the ellipse (or hyperbolic) focal length F1F2 as the base and the vertex P as ellipse (or hyperbola), and proficiency in the nature of the focus triangle is of great significance for cultivating the ability of innovation and problem solving. The focus of the example hyperbolic x29-y216 = 1 is F1, F2, and the point P is on a hyperbola. If PF1⊥PF2, then the distance from point P to the x-axis is 0. Analysis P (x0, y0) Is the distance from the point P to the x axis, so only the ordinate of the point P can be obtained Solution 1 (Auxiliary Circle Method) Auxiliary circle with the focal points F1, F2 as the diameter is constructed. From the knowledge of the circle, Circle, then F1PF2 is a right-angled triangle; if point P is within a circle, F1PF2 is an obtuse triangle; if point P is outside a circle, F1PF2 is an acute-angled triangle.