is it C?

if it is, then I will post the explanation.

x^2+y^2=100

(x+y)^2-2xy=100
(x+y)^2=100+2xy
if we can find the max value of xy, we are done
a number property:
a+b=x
ab will be max when a=b=x/2
max value of x^2*Y^2=50*50
max value of xy=50
max val of (x+y)^2=100+2*50=200
max val of x+y=sqrt(200)
answer C

Hmm. Obviously x and y can be non integers since nothing has been specified. So I just tried a bunch of combinations and found that C fits because x^2 + y^2 can be written as 50 + 50 which is (7.something)^2 + (7.something)^2 and so x = 7.smthng and y is 7.smthng and their sum is 14.smthng which is greater than 14 however no combination will yield values greater than 19 so I chose C.

Yes the answer is C