已知椭圆x2/a2+y2/b2=1(a>b>0)经过点A(2,3),焦距为4,M为右顶点,过右焦点F的直线l与椭圆于A,B两点,直线AM,BM与x=8分别交于P,Q两点,(P,Q两点不重合).(1)求椭圆的标准方程.(2)求证向量FP*向量FQ=0.
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![已知椭圆x2/a2+y2/b2=1(a>b>0)经过点A(2,3),焦距为4,M为右顶点,过右焦点F的直线l与椭圆于A,B两点,直线AM,BM与x=8分别交于P,Q两点,(P,Q两点不重合).(1)求椭圆的标准方程.(2)求证向量FP*向量FQ=0.](/uploads/image/z/10149197-5-7.jpg?t=%E5%B7%B2%E7%9F%A5%E6%A4%AD%E5%9C%86x2%2Fa2%2By2%2Fb2%3D1%28a%3Eb%3E0%29%E7%BB%8F%E8%BF%87%E7%82%B9A%282%2C3%29%2C%E7%84%A6%E8%B7%9D%E4%B8%BA4%2CM%E4%B8%BA%E5%8F%B3%E9%A1%B6%E7%82%B9%2C%E8%BF%87%E5%8F%B3%E7%84%A6%E7%82%B9F%E7%9A%84%E7%9B%B4%E7%BA%BFl%E4%B8%8E%E6%A4%AD%E5%9C%86%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E7%9B%B4%E7%BA%BFAM%2CBM%E4%B8%8Ex%3D8%E5%88%86%E5%88%AB%E4%BA%A4%E4%BA%8EP%2CQ%E4%B8%A4%E7%82%B9%2C%EF%BC%88P%2CQ%E4%B8%A4%E7%82%B9%E4%B8%8D%E9%87%8D%E5%90%88%EF%BC%89.%EF%BC%881%EF%BC%89%E6%B1%82%E6%A4%AD%E5%9C%86%E7%9A%84%E6%A0%87%E5%87%86%E6%96%B9%E7%A8%8B.%EF%BC%882%EF%BC%89%E6%B1%82%E8%AF%81%E5%90%91%E9%87%8FFP%2A%E5%90%91%E9%87%8FFQ%3D0.)
已知椭圆x2/a2+y2/b2=1(a>b>0)经过点A(2,3),焦距为4,M为右顶点,过右焦点F的直线l与椭圆于A,B两点,直线AM,BM与x=8分别交于P,Q两点,(P,Q两点不重合).(1)求椭圆的标准方程.(2)求证向量FP*向量FQ=0.
已知椭圆x2/a2+y2/b2=1(a>b>0)经过点A(2,3),焦距为4,M为右顶点,过右焦点F的直线l与椭圆于A,B两点,直线AM,BM与x=8分别交于P,Q两点,(P,Q两点不重合).(1)求椭圆的标准方程.(2)求证向量FP*向量FQ=0.
已知椭圆x2/a2+y2/b2=1(a>b>0)经过点A(2,3),焦距为4,M为右顶点,过右焦点F的直线l与椭圆于A,B两点,直线AM,BM与x=8分别交于P,Q两点,(P,Q两点不重合).(1)求椭圆的标准方程.(2)求证向量FP*向量FQ=0.
(1)
椭圆x2/a2+y2/b2=1,
焦距为4,2c=4,c=22为F'(-2,0),F(2,0)
∵点A(2,3)在椭圆上,根据椭圆定义
2a=|PF'|+|PF|=√[(4²+3²)+√(0²+3²)=8
∴a=4,b=√(a²-c²)=√12=2√3
∴椭圆的标准方程为:
x²/16+y²/12=1
(2)
l过F(2,0),设直线l:x=ty+2
x=ty+2与 x²/16+y²/12=1联立消去x
得:3(ty+2)²+4y²-48=0
即(3t²+4)y²+12ty-36=0
设A(x1,y1),B(x2,y2)
根据韦达定理:
y1+y2=-12t/(3t²+4),y1y2=-36/(3t²+4)
椭圆右顶点M(4,0),设P(8,m),Q(8,n)
根据AF与FP斜率相等,
∴m/4=y1/(x1-4),m=4y1/(x1-4)
同理:n=4y2/(x2-4)
其中x1-4=ty1-2,x2-4=ty2-2
向量FP=(6,m),向量FQ=(6,n)
∴向量FP●向量FQ
=36+mn
=36+4y1/(x1-4)*4y2/(x2-4)
=36+16y1y2/[(ty1-2)(ty2-2)]
=36+16y1y2/[t²y1y2-2t(y1+y2)+4]
=36-[16×36/(3t²+4)]/[-36t²/(3t²+4)+24t²/(3t²+4)+4]
=36-16×36/[-12t²+12t²+16]
=36-16×36/16
=36-36
=0