The equation of the parabola whose focus is the point (0,0) and the tangent at the vertex is x−y+1=0 is

Tangent at the vertex is x−y+1=0-----(1)Therefore, the equation of the axis of the parabola is X+Y=0------(2) Now, solving (1) and (2), we get A≡(−1/2,1/2) .  Therefore, Z is (−1,1). (∵A is midpoint of OZ) Now, the directrix is X-Y+K=0 But this passes through Z(−1,1). Therefore, K=0 So, the directrix is x−y+2=0Therefore, by definition, the equation of the parabola isgiven byOP=PM or OP2=PM2x−y+222=x2+y2 or  (x−y+2)2=2x2+2y2 or  x2+y2+4−2xy+4x−4y=2x2+2y2 or  x2+y2+2xy−4x+4y−4=0