Shady01, your first line is wrong.
If you were to go down the z = x + iy path, using modulus definitions:
sqrt(c) + sqrt(d) = 10
You can't just square both sides to get c + d = 100.
In fact, the easiest way to do this is to rearrange as sqrt(c) = 10 - sqrt(d) and then square both sides.
Regarding the geometric method, as people said, you're supposed to know that is the locus of an ellipse and relate it to the equation PS + PS' = 2a.
Then you can find S(1,3), S'(9,3) and a = 5 really quickly.
Then, if you were to draw a diagram, you could find the value of b = 3 through some nifty Pythagoras or some other method.