Question

请问数学大神第12题怎么写？

12

Answer 1

连接A、B，AB：

y=kx+b k=(y₂-y₁)/(x₂-x₁),b=(x₂y₁-x₁y₂)/(x₂-x₁)

x²+(kx+b)²=4→(1+k²)x²+2kbx+b²-4=0

x₁·x₂=(b²-4)/(1+k²)

y₁·y₂=(kx₁+b)(kx₂+b)=(b²-4)k²/(1+k²)+bk(x₁+x₂)+b²

       =(b²-4k²)/(1+k²)

x₁·x₂+y₁·y₂=(b²-4)/(1+k²)+(b²-4k²)/(1+k²)=-2

b²=k²+1→b=±√(1+k²)

x₁+x₂+y₁+y₂=x₁+x₂+kx₁+b+kx₂+b

=(1+k)(x₁+x₂)+2b

=(1+k)(-2kb)/(1+k²)+2b

=2b-(2k+2k²)/b

可求得k=-1(b=-√2)时，取得最小值=-2√2