If the complex number z satisfies the condition ∣∣∣z−12z∣∣∣=11, then the maximum distance from the origin to the point representing z in the Argand plane is
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Solution
Given : 11=∣∣∣z−12z∣∣∣
We know that |a−b|≥|a|−|b|, so ∣∣∣z−12z∣∣∣≥|z|−12|z|⇒11≥|z|−12|z|⇒|z|2−11|z|−12≤0⇒(|z|+1)(|z|−12)≤0⇒−1≤|z|≤12⇒0≤|z|≤12(∵|z|≥0)⇒|z−(0+0i)|≤12
Hence, the maximum distance of z from the origin is 12.