Question

若直线2ax-by+2=0(a,b大于0)始终平分圆x^2+y^2-2x-4y-6=0的周长,则2

Answer 1

x²+y²-2x-4y-6=0

(x-1)²+(y-2)²=11

因为直线2ax+by-2=0平分圆

所以直线过圆的圆心

圆的圆心是(1,2)

将(1,2)代入直线2ax+by-2=0

则2a+2b-2=0

则a+b=1

则2=a+b,

2/a+1/b

=2(a+b)/a+(a+b)/b

=2+(2b/a)+(a/b)+1

=3+(2b/a)+(a/b)

根据重要不等于a²+b²≥2ab

则(2b/a)+(a/b)≥2√2

则3+(2b/a)+(a/b)≥3+2√2

则最小值是3+2√2