已知圆C与直线x-y=0及x-y-4=0都相切，圆心在直线x+y=0上，则圆C的方程为（    ）

A：${\left(x+1\right)}^2+{\left(y-1\right)}^2=2$

B：${\left(x-1\right)}^2+{\left(y+1\right)}^2=2$

C：${\left(x+1\right)}^2+{\left(y-1\right)}^2=8$

D：${\left(x-1\right)}^2+{\left(y+1\right)}^2=8$