This problem is not that difficult . However, its lengthy is troublesome.

To solve it faster you have two ways

1) Or manipulate the equation $\frac{y+4}{y-4}=\frac{15}{(x-4)(y-2)}=\frac{x-2}{y-2}$ and essentially you will end up with the same equation $(x-a)^2+(y-b)^2=r^2$

2) Or use the answer choices and see what is going on

In the first case after manipulation, we will have

$(x-3)^2+(y+1)^2=25$

Where a=3, b=-1 and r=5

Choice D is correct