过点B(o,-b),作椭圆x^2/a^2+y^2/b^2=1(a>b>0) 求这些弦的最大值
1个回答
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)2
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