Question

若直线2ax-by+2=0(a>0,b>0)平分圆x²+y²+2x-4y+1=0,则1/a+1/b的最小值

Answer 1

x²+y²+2x-4y+1
=
(x+1)²+(y-2)²-4
所以圆心是(-1,
2)
2ax-by+2=0平分圆，
和圆相交截得线段是直径。所以2ax-by+2=0
过圆心。
2a(-1)
-2b
+2
=0,
即a+b
=1
a+b
>=
2√ab,
2√ab<=1
ab
<=
1/4
1/ab
>=
4
1/a
+
1/b
=
(b+a)/ab
=
1/ab
>=4