Question

若直线ax+by+1=0,过圆x^2+y^2+8x+2y+1=0的圆心，则1/a+4/b的最小值

Answer 1

x^2+y^2+8x+2y+1=0
(x+4)^2+(y+1)^2-16=0
所以x^2+y^2+8x+2y+1=0的圆心为(-4,-1)
所以-4a-b+1=0
即4a+b=1
1/a+4/b=(4a+b)/ab=1/ab
ab=a(1-4a)=a-4a^2=-(2a-1/4)^2+1/16
a=1/4时,ab有最大值1/16
1/a+4/b=(4a+b)/ab=1/ab有最小值16

Answer 2

x^2+y^2+8x+2y+1=0(x+4)�0�5+(y+1)�0�5=16所以圆心为（-4，-1）带人直线方程-4a-b+1=04a+b=11=4a+b≥2倍根号(4ab)=4倍根号(ab)根号ab≤1/4ab≤1/21/(ab)≥2 1/a+4/b=(4a+b)/(ab)=1/(ab)≥2最小值为2