椭圆的中心在原点,焦点在y轴上,且焦距=14，P在椭圆上，∠F1PF2=2派/3，若△F1PF2的面积为13根号3，求方程

2c=|F1F2|=14,c=14/2=7
S△F1PF2=1/2*|F1P||F2P|sin∠F1PF2
|F1P||F2P|=2*13√3/(sin2π/3)=52
而：|F1F2|^2=|F1P|^2+|F2P|^2-2|F1P||F2P|cos∠F1PF2
于是：|F1P|^2+|F2P|^2=|F1F2|^2+2|F1P||F2P|cos∠F1PF2=196+2*52cos2π/3
|F1P|^2+|F2P|^2=144
(2a)^2=(|F1P|+|F2P|)^2=|F1P|^2+|F2P|^2+2|F1P||F2P|=248,a^2=62
b^2=a^2-c^2=62-49=13
椭圆为:x^2/13+y^2/62=1.

焦距F1F2=14,c=14/2=7
△F1PF2的面积=1/2*|F1P||F2P|sin∠F1PF2
|F1P||F2P|=2*13√3/(sin2π/3)=52
|F1P|^2+|F2P|^2=|F1F2|^2+2|F1P||F2P|cos∠F1PF2
=14^2+2*52*cos2π/3
=196-52
=144
(|F1P|+|F2P|)^2=|F1P|^2+|F2P|^2+2|F1P||F2P|
=144+2*52
=248
|F1P|+|F2P|=2√62
所以，2a^2/c=2√62
a^2=√62*c=7√62
b^2=a^2+c^2=7√62+49
所以，椭圆的标准方程：x^2/7√62+y^2/(7√62+49)=1