Suppose that the complex number \( z \) lies on the curve such that \( \frac{z-4}{z-2 i} \) is p...

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Suppose that the complex number \( z \) lies on the curve such that \( \frac{z-4}{z-2 i} \) is purely imaginary. If the complex number \( z_{1} \) represents the mid-point of chord \( O A \) of this curve, \( O \) being the origin, then \( z_{1} \) necessarily satisfy
(a) \( \frac{z_{1}-2}{z_{1}-i}=i k, k \in R-\{0\} \)
(b) \( \frac{z_{1}}{z_{1}-2-i}=i k, k \in R-\{0\} \)
(c) \( \frac{z_{1}-2}{2 z_{1}-i}=k, k \in R-\{0\} \)
(d) \( \left|z_{1}\right|=\frac{\sqrt{5}}{2} \)
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