Question:
Given that,
$$|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16
$$
Given these conditions, we need to find the difference between the maximum and minimum values of $|z_1^3 + z_2^3|$.
I undersdand this problem requires careful manipulation of complex numbers. But i couldn't manipulate it to a reasonable extent.
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We're asked to find the difference between the maximum and minimum values of $|z_1^3 + z_2^3|$. Using De Moivre's theorem, we have:
Given hint in the answer key:
To find the extreme values, we differentiate with respect to $\theta_1 – \theta_2$, equate it to zero, and solve for $\theta_1 – \theta_2$. We then substitute this value back to get the maximum and minimum values of $|z_1^3 + z_2^3|$.
This seems way lengthy and unreasonable, is there any elegant or better approach than this?
Best Answer
(The idea similar as in John Bentin's answer .)
With $a=z_1 + z_2$ and $b = z_1^2 + z_2^2$ is $$ z_1^3 + z_2^3 = -\frac 12 a (a^2 - 3b) $$ and therefore $$ |z_1^3 + z_2^3| = \frac 12 |a| \cdot |a^2 - 3b| = 10 |a^2 -3b| \\ \begin{cases} \le 10 (|a|^2 + 3|b|) = 4480 \\ \ge 10 (|a|^2 - 3|b|) = 3520 \end{cases} $$ by the triangle inequality. The bounds are sharp:
Both cases are possible because the system $$z_1+z_2=a, z_1^2+z_2^2=b$$ has solutions for any pair $(a, b)$ of complex numbers.
Therefore the desired difference is $4480-3520 = 960$.