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The system of equations \(\begin{align*} |z - 2 - 2i| &= \sqrt{23}, \\ |z - 8 - 5i| &= \sqrt{38} \end{align*}\)
has two solutions \(z_1\) and \(z_2\) in complex numbers. Find \((z_1 + z_2)/2\).

 Dec 15, 2019
 #1
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Express z_1 and z_2 in rectangular form, and solve the quadratic equations!  You get z_1 = 5 + 2/7*sqrt(10) + i(8 - 3/7*sqrt(10)) and z_2 = 5 - 2/7*sqrt(10) + i(8 + 3/7*sqrt(10)), so (z_1 + z_2)/2 = 5 + 8i.

 Dec 15, 2019
 #2
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I get a somewhat different answer from Guest #1:

 

 Dec 15, 2019

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