If z - 4z = 2 then the greatest value of z is

z=z-4z+4z ⇒z≤z-4z+4z ∴z≤2+4z⇒z2-2z≤4 ⇒z2-2z+1≤5 ∴z-12≤5 ⇒z-1≤5 ∴z≤5+1 greatest value of z=5+1

If z - 4z = 2 then the greatest value of z is

z=z-4z+4z ⇒z≤z-4z+4z ∴z≤2+4z⇒z2-2z≤4 ⇒z2-2z+1≤5 ∴z-12≤5 ⇒z-1≤5 ∴z≤5+1 greatest value of z=5+1

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If $|z - \frac{4}{z}|$ = 2 then the greatest value of $|z|$ is

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$\sqrt{5} - 1$

$1 - \sqrt{5}$

$\sqrt{5}$

$\sqrt{5} + 1$

answer is B.

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Detailed Solution

$|z| = |(z - \frac{4}{z}) + \frac{4}{z}| \Rightarrow |z| \leq |z - \frac{4}{z}| + |\frac{4}{z}| ∴ |z| \leq 2 + \frac{4}{|z|} \Rightarrow {|z|}^{2} - 2 |z| \leq 4 \Rightarrow {|z|}^{2} - 2 |z| + 1 \leq 5 ∴ {(|z| - 1)}^{2} \leq 5 \Rightarrow |z| - 1 \leq \sqrt{5} ∴ |z| \leq \sqrt{5} + 1 greatest value of |z| = \sqrt{5} + 1$