Greatest value

I don't really see a shorter way. The shortest way would be to know properties of numbers so well that you do not need to plug them in, you simply think of them in terms of the answer choices. I am certainly not at that level. I still had to plug in. Curious to see what others think.

I would approach is like this:

You have X^2 everywhere, so you can just ignore it and focus on the Y part of the problem.

You should just think that Y has a "good" (i.e. the final answer will be greater) effect when it is in the denominator and a "bad" effect when it is in the numerator.
--> Keep only C and D

Since Y has a good effect in the denominator, it will be better to have it inside the square. (it is better to divide by 1/4 (=multiply by 4) than to divide by 1/2 (=multiply by 2)
--> Choice C

Is that the correct answer?

Magellan,

Rethink this by inserting saome values:
Say, x=-10 and y = 1/2
Then, (x/y)^2 =100*4=400
And in D: 100*2=200.
So, C is the greatest.

I said C as well