Let the co-ordinates of P are (x,y) According to given condition (x−1)2+(y−1)2=
(x+y+2)2
2
⇒2(x2+1−2x)+2(y2+1−2y) =x2+y2+4+2xy+4y+4x ⇒x2+y2−2xy−8x−8y=0 . . . (i0 On comparing with ax2+by2+2hxy+2gx+2fy+c=0, we get a=1,b=1,h=−1,g=−4,f=−4,c=0 ∴abc+2fgh−af2−bg2−ch2 =1.1.0+2(−4)(−4)(−1)−1(−4)2−1(−4)2−0 =0−32−16−16=−64≠0 and h2−ab=1−1=0 Since ∆≠0 and h2=ab, therefore locus of P is a parabola.