(x-h)^2+(y-k)^2=r^2 where (h,k) is the circle's centre and r is the radius.
Substitute the points:
h^2+(5-k)^2=r^2=(5+h)^2+k^2
h^2+25-10k+k^2=25+10h+h^2+k^2
-10k=10h, k=-h
(3-h)^2+(4-k)^2=(3+h)^2+(4+k)^2
9-6h+h^2+16-8k+k^2=9+6h+h^2+16+8k+k^2
-12h-16k=0, 3h+4k=0, and k=-h so h and k are zero, therefore r=5 (x^2+y^2=25).