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椭圆、双曲线上不与两个焦点共线的任意一点与两个焦点组成的三角形叫做椭圆、双曲线的焦点三角形。涉及焦点三角形面积的试题多次出现在高考题中,直接解答一般较复杂,若我们能合理而又灵活地运用椭圆、双曲线的焦点三角形的面积公式,在解决这类有关问题时,可避免冗长的推理和运算,大大降低难度,简化运算。一、课本例题若P是椭圆x2100+y264=1上的一点,F1、F2是其焦点,且,求△∠F1PF2=60°,求△∠F1PF2的面积解法一:在椭圆x2100+y264=1中,a=10,b=8,c=6,而θ=60°记PF1=r1,PF2=r2∵点P在椭圆上,∴由椭圆的第一定义得:r1+r2=2a=20在△F1PF2中,由余弦定理得:r12+r22-2r1r2cosθ=(2c)2配方,得:(r1+r2)2-3r1r2=144256
Ellipse, hyperbolic not collinear with the two points at any one point and the two focal points composed of triangles called ovals, hyperbolic focus triangle. Questions involving the area of the focus triangle appear many times in the entrance exam questions, and the direct solutions are generally more complex. If we can reasonably and flexibly use the area formula of the focal triangle of ellipses and hyperbolic curves to avoid such problems Lengthy reasoning and computing, greatly reducing the difficulty and simplify computing. First, the textbook example If P is a point on the elliptic x2100 + y264 = 1, F1, F2 is the focus, and, find △ ∠ F1PF2 = 60 °, find △ ∠ F1PF2 area Solution 1: In the elliptic x2100 + y264 = 1 = 1, b = 8, c = 6, and θ = 60 °. Note that Pf1 = r1, Pf2 = r2. Point P is on the ellipse and ∴ is defined by the first ellipse: r1 + r2 = 2a = 20 In △ F1PF2, by the cosine theorem: r12 + r22-2r1r2cosθ = (2c) 2 formula, was: (r1 + r2) 2-3r1r2 = 144256