Consider the set of complex number z satisfying $| 1 + z + z^{2} | = 4$ . The maximum value of the imaginary part of z can be written in the form $\sqrt{\frac{m}{n}}$ , where m and n are relatively prime positive integers. What is $m + n ?$

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Detailed Solution

Let $ω = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ be a primitive $3^{rd}$ root of unity. Then
$4 = | z^{2} + z + 1 | = | (z - ω) (z - \bar{ω}) | \geq | im (z - ω) im (z - \bar{ω}) | = | (im (z) - \frac{\sqrt{3}}{2}) (im (z) + \frac{\sqrt{3}}{2}) |$
$= | im {(z)}^{2} - \frac{3}{4} |$ implying $im (z) \leq \sqrt{4 + \frac{3}{4}} = \frac{\sqrt{19}}{2} (B)$ . Equality occurs when $z = - \frac{1}{2} + \frac{\sqrt{19}}{2} i$