This is a multiple-choice problem with 4 potential answers.

So, it must be true that the answer is the same no matter what x and y are,

as long as |z| = |x+iy| = 8

That's because your teacher couldn't have given you only 4 numerical choices if
there could be different values for different values of x and y.
 
So, instead of bothering to prove it for all values of x and y, just choose this
easy case:

x=8, y=0, then z = 8+0i = 8-0i = conj(8+0i) = 8, then

|(conj(z)+48)/(3z + 4)| = |8+48)/[(3)(8) + 4]| = |56/28| = |2| = 2.

So the answer is 2. 

Note: if your problem had been stated this way: 

if |z| = 8 then prove that the value of |(conj(z)+48)/(3z + 4)| = 2,

then there would be a lot more work.  So be on the lookout for multiple
choice questions like this with variables which have no variables in the
choices.  For if you run into such a problem, just substitute easy numbers
for the variables and whatever you get will bound to be the correct answer.
You'll save lots of time.

Edwin