# Complex with Geometry

Algebra Level 5

Consider the circle $$|z-5i| = 3$$ , and two points $$z_1 , z_2$$ on it such that $$|z_1|<|z_2|$$ , $$\text{arg}(z_1) = \text{arg}(z_2) = \frac{\pi}{3}$$. A tangent is drawn at $$z_2$$ to the circle, which cuts the real axis at $$z_3$$. What is the magnitude of $$z_3$$?