From your graph, what do you know about the imaginary part of points in C1 ? If the point is also in C2, you have its argument. Use these two pieces of information to find the real part of your point.
Just an additional point of view. For future reference, try to visualize the geometry beforehand using the geometric meaning of the equations. For example, your first equation states that |z - 0| = |z - 4i|. This means that the distance between z and 0 is the same as the distance between z and 4i. This is describing the horizontal line of points with imaginary part 2 (right between 0 and 4i).
The second equation was covered by the previous poster as a ray. You can then find the intersection point by using basic trigonometry, as the ray, horizontal line, and imaginary axis form a right triangle.
While you can work it out algebraically, it is often much faster to see the full geometric picture first, using the concepts of distance and angle in the complex plane as analogues to modulus and argument, respectively. Of course, this only helps if you have a solid grounding in plane geometry as well (ie., you should know the conics as geometric descriptions of loci, not just their Cartesian equations).