关于圆锥曲线方程问题1.若过点B(0,-b)作椭圆x^2/a^2+y^2/b^2=1(a>b>0)的弦,求这些弦长的最大值.2.已知A,B,D三点不在一条直线上,且A(-2,0),B(2,0),|向量AD|=2,向量AE=1/2*(向量AB+向量AD) (1)
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![关于圆锥曲线方程问题1.若过点B(0,-b)作椭圆x^2/a^2+y^2/b^2=1(a>b>0)的弦,求这些弦长的最大值.2.已知A,B,D三点不在一条直线上,且A(-2,0),B(2,0),|向量AD|=2,向量AE=1/2*(向量AB+向量AD) (1)](/uploads/image/z/5731682-50-2.jpg?t=%E5%85%B3%E4%BA%8E%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF%E6%96%B9%E7%A8%8B%E9%97%AE%E9%A2%981.%E8%8B%A5%E8%BF%87%E7%82%B9B%280%2C%EF%BC%8Db%EF%BC%89%E4%BD%9C%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%E7%9A%84%E5%BC%A6%2C%E6%B1%82%E8%BF%99%E4%BA%9B%E5%BC%A6%E9%95%BF%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC.2.%E5%B7%B2%E7%9F%A5A%2CB%2CD%E4%B8%89%E7%82%B9%E4%B8%8D%E5%9C%A8%E4%B8%80%E6%9D%A1%E7%9B%B4%E7%BA%BF%E4%B8%8A%2C%E4%B8%94A%EF%BC%88-2%2C0%EF%BC%89%2CB%EF%BC%882%2C0%EF%BC%89%2C%EF%BD%9C%E5%90%91%E9%87%8FAD%EF%BD%9C%3D2%2C%E5%90%91%E9%87%8FAE%3D1%2F2%2A%EF%BC%88%E5%90%91%E9%87%8FAB%2B%E5%90%91%E9%87%8FAD%EF%BC%89+%EF%BC%881%EF%BC%89)
关于圆锥曲线方程问题1.若过点B(0,-b)作椭圆x^2/a^2+y^2/b^2=1(a>b>0)的弦,求这些弦长的最大值.2.已知A,B,D三点不在一条直线上,且A(-2,0),B(2,0),|向量AD|=2,向量AE=1/2*(向量AB+向量AD) (1)
关于圆锥曲线方程问题
1.若过点B(0,-b)作椭圆x^2/a^2+y^2/b^2=1(a>b>0)的弦,求这些弦长的最大值.
2.已知A,B,D三点不在一条直线上,且A(-2,0),B(2,0),|向量AD|=2,向量AE=1/2*(向量AB+向量AD)
(1)求E点的轨迹方程;
我已求出是x^2+y^2=1(y不等于0)
(2)过A作直线交以A、B为焦点的椭圆于MN的中点到y轴的距离为4/5,且直线MN与E点的轨迹相切,求椭圆方程.
关于圆锥曲线方程问题1.若过点B(0,-b)作椭圆x^2/a^2+y^2/b^2=1(a>b>0)的弦,求这些弦长的最大值.2.已知A,B,D三点不在一条直线上,且A(-2,0),B(2,0),|向量AD|=2,向量AE=1/2*(向量AB+向量AD) (1)
1.a>b>0
过点B(o,-b)的弦:
y=kx-b
x^2/a^2+y^2/b^2=1
b^2*x^2+a^2*y^2=(ab)^2
b^2*x^2+a^2*(kx-b)^2=(ab)^2
(b^2+a^2*k^2)x^2-2bka^2*x=0
x1+x2=2bka^2/(b^2+a^2*k^2),x1*x2=0
(x1-x2)^2=(x1+x2)^2-4x1*x2=[2bka^2/(b^2+a^2*k^2)]^2
(y1-y2)^2=k^2*(x1-x2)^2
弦L^2=(x1-x2)^2+(y1-y2)^2=(1+k^2)*[2bka^2/(b^2+a^2*k^2)]^2
a^4*(4b^2-L^2)k^4+(ab)^2*(4a^2-2L^2)k^2-L^2*b^4=0
[(ab)^2*(4a^2-2L^2)]^2-4a^4*(4b^2-L^2)*(-L^2*b^4)≥0
L^2≤a^4/(a^2-b^2)
a^4/(a^2-b^2)=a^4/c^2=(a^2/c)^2=(a*e)^2
L≤a^2/c=a*e
如果弦在Y轴上,则弦长=2b
答:
(1)2b>a^2/c,这些弦的最大值=2b
(2)2b