y=-x+ 1

x= 1-y

Then: y? =2p( 1-y)

y？ +2py-2p=0

y 1+y2=-2p，？ y 1y2=-2p

x 1x 2 =( 1-y 1)( 1-y2)= 1-(y 1+y2)+y 1 y2 = 1+2p-2p = 1

The circle with the diameter of the chord AB just passes through the origin.

OA⊥OB

x 1x2+y 1y2=0

x 1x 2+y 1 y2 = 1-2p = 0

p= 1/2

The equation of parabola is: y? =x？