(1)椭圆过点(√6,1),
∴6/a^2+1/b^2=1.
离心率c/a=1/√2,∴a^2=2c^2,b^2=c^2,代入上式得4/c^2=1,c^2=4,a^2=8,
∴椭圆的标准方程是x^2/8+y^2/4=1.
(2)设A(2√2cosu,2sinu),B(2√2cosv,2sinv),则
向量PA*PB=(2√2cosu-√6,2sinu-1)*(2√2cosv-√6,2sinv-1)
=(2√2cosu-√6)(2√2cosv-√6)+(2sinu-1)(2sinv-1)
=8cosucosv-4√3(cosu+cosv)+6+4sinusinv-2(sinu+sinv)+1=-2,
∴8cosucosv+4sinusinv-4√3(cosu+cosv)-2(sinu+sinv)+9=0,①
AB的斜率=(sinu-sinv)/[√2(cosu-cosv)],
AB的方程是y-2sinu=(sinu-sinv)(x-2√2cosu)/[√2(cosu-cosv)],
设AB过定点(m,n),则√2(cosu-cosv)(n-2sinu)=(sinu-sinv)(m-2√2cosu),
∴√2(cosu-cosv)n-(sinu-sinv)m+2√2(sinucosv-cosusinv)=0.②
不存在常数m,n,使得方程①、②同解,
∴AB不过定点.