O(0,0) lies on the circle.

general form of any point on this circle is given by (-g+r(cosØ),-f+r(sinØ)) hence here it is P(1+1(cosØ),1(sinØ)).

thus for mid pt. of chord OP:

x= (0+1+1cosØ)/2 =>(1+cosØ)/2    (1)

y=sinØ/2;     (2)

from (1) and (2), (cosØ)^2= (2x-1)^2,   (3)

                          (sinØ)^2=(2y)^2;     (4)

adding (3) and (4)

     sqr(2x-1)+sqr(2y)=1,

 or x^2 +y^2-x=0 ,which is the required eq.