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 \(z_3 \)?

Give your answer correct to three decimal places.

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