Question

一道高中数学题，帮忙做一下，谢谢了

Answer 1

c^2/a^2 = 1/2
c^2=b^2=a^2/2
x=c y=根号2/2

1/2+1/2b^2=1
=>b^2=1 a^2=2 C的方程x^2/2+y^2=1

y=kx+b
x^2+2y^2=2
(1+2k^2)x^2+4kbx+2b^2-2=0
x1 = -2kb/(1+2k^2)

仅有一个根，b^2-4ac=0
x1 = -b/2a

可推出
且 16k^2b^2-4(1+2k^2)(2b^2-2)=0
2k^2b^2-(1+2k^2)(b^2-1)=0
2k^2b^2-b^2+1-2k^2b^2+2k^2=0
b^2 = 2k^2+1

圆的可以由
y=kx+b
x^2+y^2=r^2
(1+k^2)x^2+2kbx+b^2-r^2=0
x2=-kb/(1+k^2)

|AB|^2 = (x2-x1)^2+(y2-y1)^2=(1+k^2)(x2-x1)^2
=1/(2k^2+1/k^2+3)

2k^2+1/k^2+3>=3+2根号2
当2k^2=1/k^2 k^4 = 1/2时取等号
所以|AB|^2<=1/(3+2根号2)

|AB| <=[1/(3+2根号2)]^(1/2) = 1/(根号2+1) = 根号2-1

Answer 2

高中毕业8年了，发现自己越来越笨

Answer 3

太麻烦了

Answer 4

太难了