Question

过圆x2+y2=1上一点P作圆的切线与X轴和Y轴分别交于A,B两点,O是坐标原点,则OA+8OB的最小值是＿

Answer 1

设OA=X,OB=Y

OA+8OB=X+8Y>=2√(X*8Y) 当 X=8Y 时有最小值2√(X*8Y).

连接D(0,1),C(8,0).得两直角边为1,8的三角形,其斜边长√65,其斜边上的高OE=8/√65,

作该圆切线平行于BD,切点P.切线与X轴和Y轴分别交于A,B两点.有

OB/OD=OA/OC=OP/OE

Y/1=X/8=1/(8/√65),X=√65,Y=√65/8

OA+8OB=X+8Y>=2√(X*8Y)=2√65

OA+8OB的最小值是＿2√65