Given a and b in R for the system are unique solution :
$\displaystyle
(x-1)^{2}+(y+1)^{2}=b
$
$\displaystyle
y = x^{2}+(2a+1)x+a^{2}
$
Do you mean: find values of $\displaystyle a$ and $\displaystyle b$ such that the system has unique solution?
Fernando Revilla