(1)
圆1:(x-1)^2+y^2=2 ①
圆心O(1,0),
以OP为直径作圆2:(x+1/2)^2+(y+1/2)^2=5/2 ②
设圆1与圆2交点为Q1,Q2
显然PQ1,PQ2与圆1相切.PQ1,PQ2就是待求切线.
①②联立,可得:
3x-3/4+y+1/4=3x+y-1/2=1/2 => y=1-3x
(x-1)^2+(1-3x)^2=2
10x^2-8x=0
x=0或x=4/5
x=0时,y=1,切线方程为:(y-1)/x=-2/(-2)=1,y=1+x
x=4/5时,y= -7/5,切线方程为:(y+1)/(x+2)=-2/5/(14/5)=-1/7,
x+7y+9=0
综上所述,切线方程为y=1+x和x+7y+9=0
(2) 设夹角为s
sin(s/2)=2^0.5 / 10^0.5 = 1/5^0.5
s = 2 arcsin(1/5^0.5)=arcsin(4/5)