Source : Complex Variables by Ablowitz and Fokas
Q : Show that a circle in the z- plane corresponds to a circle on the sphere.
Sphere : $X^2+Y^2+(Z-1)^2=1$
Attempt : x,y are in z-plane (complex)
Transformations Used $X=\frac{4x}{x^2+y^2+4},Y=\frac{4y}{x^2+y^2+4},Z=\frac{2(x^2+y^2)}{|z|^2+4}$
$x=\frac{2X}{2-Z},y=\frac{2Y}{2-Z}$
Let the equation of circle on z-plane be $A(x^2+y^2)+Bx+Cy+D=0$
After substituting x,y in above equation and using Equation of sphere. I got
$2BX+2CY+Z(A-D)+2D=0$
This is equation of plane; why I am not able to get equation of circle?
Where's my mistake ?