When origin is shifted to (2,b) by translation of axes, the coordinates of the point (a,4) have changed to (6,8) ∴X=x−2⇒Y=y−b For point (a,4) transforming to (6,8) 6=a−2⇒a=8 8=4−b⇒b=−4 So, the point (a,b) is (8,−4). Now, when origin is shifted to (a,b) ⇒‌‌X=x−8⇒Y=y+4 and transformed equation of x2+4xy+y2=0 is ‌(X+8)2+4(X+8)(Y−4)+(Y−4)2=0 ‌X2+Y2+4XY+24Y−48=0 On comparing with X2+2HXY+Y2+2GX+2FY+C=0 we get, G=0,H=2,F=12,C=−48 ∴2H(G+F)=2×2(0+12)=48=−C
