椭圆E:x^2/a^2+y^2/b^2=1(a>b>0)的左右焦点分别为F1,F2点,A(4,m)在椭圆E上,且向量AF2*向量F1F2=0,点D(2,0)到直线F1A的距离DH=18/5(1)椭圆E的方程(2)设点P为椭圆E上任意一点,求向量PF1*向量PD的取值范
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![椭圆E:x^2/a^2+y^2/b^2=1(a>b>0)的左右焦点分别为F1,F2点,A(4,m)在椭圆E上,且向量AF2*向量F1F2=0,点D(2,0)到直线F1A的距离DH=18/5(1)椭圆E的方程(2)设点P为椭圆E上任意一点,求向量PF1*向量PD的取值范](/uploads/image/z/5185902-30-2.jpg?t=%E6%A4%AD%E5%9C%86E%3Ax%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%E7%9A%84%E5%B7%A6%E5%8F%B3%E7%84%A6%E7%82%B9%E5%88%86%E5%88%AB%E4%B8%BAF1%2CF2%E7%82%B9%2CA%284%2Cm%29%E5%9C%A8%E6%A4%AD%E5%9C%86E%E4%B8%8A%2C%E4%B8%94%E5%90%91%E9%87%8FAF2%2A%E5%90%91%E9%87%8FF1F2%3D0%2C%E7%82%B9D%EF%BC%882%2C0%EF%BC%89%E5%88%B0%E7%9B%B4%E7%BA%BFF1A%E7%9A%84%E8%B7%9D%E7%A6%BBDH%3D18%2F5%EF%BC%881%EF%BC%89%E6%A4%AD%E5%9C%86E%E7%9A%84%E6%96%B9%E7%A8%8B%EF%BC%882%EF%BC%89%E8%AE%BE%E7%82%B9P%E4%B8%BA%E6%A4%AD%E5%9C%86E%E4%B8%8A%E4%BB%BB%E6%84%8F%E4%B8%80%E7%82%B9%2C%E6%B1%82%E5%90%91%E9%87%8FPF1%2A%E5%90%91%E9%87%8FPD%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83)
椭圆E:x^2/a^2+y^2/b^2=1(a>b>0)的左右焦点分别为F1,F2点,A(4,m)在椭圆E上,且向量AF2*向量F1F2=0,点D(2,0)到直线F1A的距离DH=18/5(1)椭圆E的方程(2)设点P为椭圆E上任意一点,求向量PF1*向量PD的取值范
椭圆E:x^2/a^2+y^2/b^2=1(a>b>0)的左右焦点分别为F1,F2点,A(4,m)在椭圆E上,且向量AF2*向量F1F2=0,点D(2,0)到直线F1A的距离DH=18/5
(1)椭圆E的方程
(2)设点P为椭圆E上任意一点,求向量PF1*向量PD的取值范围
椭圆E:x^2/a^2+y^2/b^2=1(a>b>0)的左右焦点分别为F1,F2点,A(4,m)在椭圆E上,且向量AF2*向量F1F2=0,点D(2,0)到直线F1A的距离DH=18/5(1)椭圆E的方程(2)设点P为椭圆E上任意一点,求向量PF1*向量PD的取值范
(1) AF2 * F1F2 =0,所以两向量垂直,
则F2坐标为(4,0),F1坐标为(-4,0),c=4,
椭圆准线x=+/-a^2/4;
三角形F1DH相似与三角形F1AF2,则F1H/F1F2 = DH/F2A ; (1)
F1H=根号(F1D^2-DH^2)=根号(6^2-(18/5)^2)=24/5;
所以由(1)式得:(24/5)/8=(18/5)/m;得到m=6;
根据准线的性质可得:a^2/4-4=6 ,所以a=2倍的根号10;
则b=根号(a^2-c^2)=2倍的根号6;
所以椭圆E的方程为:x^2/40+y^2/24=1;
(2) 设P点坐标(x,y),设M=PF1 * PD=(x+4,y)*(x-2,y)=x^2+2x-8+y^2;
则M=x^2+2x-8+y^2=x^2+2x-8+(24-3x^2/5)
=2x^2/5+2x+16 (x大于等于-2倍的根号10,小于等于2倍的根号10)
在二次函数的对称轴x=-2.5上取的最小值Mmin=17/2;
在x=2倍的根号10时取得最大值Mmax=32+4倍的根号10.
综上:取值范围是 17/2