Question on Pareto optimality problem In an economy with two agents whose utility functions are
$$
U_A(x_1,x_2) = \alpha \cdot x_1 + x_2 \hskip 20pt U_B(y_1,y_2) = y_1 \cdot y_2.
$$

The given allocations are bundle (4,0) for A and bundle (1,5) for B.

Consider the following question


Taking into consideration the respective utilities for the bundles, we have $U_A=4\alpha$ and $U_B=5$. For this allocation to be a No Envy allocation, it has to be $4\alpha \geq 5$, which means alpha has to be greater than or equal to $\frac{5}{4}$. 

Is this the right approach to solve this problem? If its not, please find the solution and show me the steps.
As denesp mentioned, your solution is incorrect. To check whether there is envy, you have to compare the two bundles from the point of view of the same agent. So in our case we have:


For agent A: the utility of his own bundle is $4 \alpha$ and the utility of the other bundle is $\alpha+5$. So he feels no envy as long as $4\alpha \geq \alpha+5$, which is equivalent to: $\alpha \geq 5/3$.
For agent B: the utility of his own bundle is 5 and the utility of the other bundle is 0. So he never feels envy.


This means that the correct answer is (c).