Could you please show what you tried? I see nothing at all unusual about this problem.
For any "Lagrange multiplier" problem in two variables, you will get two equations of the form and .
Dividing one equation by the other gives an equation that does not involve : which, in this case, is particularly simple.
This is one form of Lagrange multipliers, that is taught in calculus courses. This question is from the optimization book of Nash & Sofer, they use some different notation and solve similar types of questions using matrices, which is confusing for me. But i see that this type of problems can be solved by just using Calculus "Lagrange multipliers", right?