Question

已知圆C:x^2+y^2+bx+ay-3=0(a,b为正实数)上任意一点关于直线l:x+y+2=0的对称点都在圆C上,则

Answer 1

(-b/2,-a/2)在直线x+y+2=0上

所以 a+b=4

1/a+3/b=(b+3a)/ab=(4+2a)/ab

=(4+2a)/(4a-a^2)

令t=4+2a a=(t-4)/2

t/{2t-8-(t-4)^2/4}=4t/{8t-32-t^2+8t-16}

=4t/(-t^2+16t-48)

=4/{-t+16-48/t}

当 t=48/t 即t=4乘以根号3 {满足a=(t-4)/2>0}时 有最小值

4/(16-8根号3)=1-根号3/2