How to derive utility possibility frontier?
Given that $u_A(x_A, y_A) = \sqrt{x_Ay_A}$, $u_B(x_B, y_B) = \sqrt{x_By_B}$ are the utility functions of A and B, and total endowment of X and Y in this pure exchange economy is $\omega_X$ and $\omega_Y$, utility possibility frontier (UPF) is the set of all utility pairs $(\mu_A, \mu_B)$ such that the there exist a Pareto optimal allocation $((x_A, y_A), (x_B, y_B))$ with the property: $\mu_A = u_A(x_A, y_A)$ and $\mu_B = u_B(x_B, y_B)$. So to find UPF, we will use the following conditions:


Optimality:


$\displaystyle \frac{y_A}{x_A} =\frac{y_B}{x_B}$ 


Feasibility:


$x_A + x_B = \omega_X$ and $y_A + y_B = \omega_Y$


Utility:


$\mu_A = \sqrt{x_Ay_A}$ and $\mu_B = \sqrt{x_By_B}$

Solving the above system of equations we get the UPF as:
$\mu_A + \mu_B = \sqrt{\omega_X\omega_Y}$.