过点M(1,0)且斜率是1的直线方程是y=x-1
代入到抛物线中有:x^2-2x+1=2px
x^2-(2+2p)x+1=0
x1+x2=2+2p,x1x2=1
AB=根号(1+k^2)*|x1-x2|=根号(1+1)*根号[(x1+x2)^2-4x2x1]=根号2*根号[4+8p+4p^2-4]=4
2(8p+4p^2)=16
p^2+2p-2=0
(p+1)^2=3
p=(+/-)根号3-1
(2)p=2,y^2=4x,焦点坐标是(1,0),即AB过焦点,则有AB的长=x1+x2+p=2+2p+p=2+3p=2+6=8
原点到AB的距离是d=|-1|/根号2=根号2/2
故S(OAB)=1/2AB*d=1/2*8*根号2/2=2根号2