How to derive utility possibility frontier? I'm trying to derive the utility possibility frontier of a economy whose consumption contract curve is $$y_A = \frac {y} {x} x_A$$ and $$y_B = \frac {y} {x} x_B$$where $x_A + x_B = x$ and $y_A + y_B= y$. 

$$U_i = \sqrt {x_iy_i}$$

While deriving the UPF. they have added up $U_A$ and $U_B$, and I don't understand why. Why do we add up utilities while deriving UPF?

(I gave all the details of the question because I'm not sure if they added the utilities only because of the peculiarities of the question or if its an algorithm. )
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:
$\mu_A + \mu_B = \sqrt{\omega_X\omega_Y}$.