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The least value of |Z - 3 - 4i|² + |Z + 2 - 7i|² + |Z - 5 + 2i|² occurs when Z equals (where Z is a complex number):

1. 1 + 3i
2. 3 + 3i
3. 3 + 4i
4. None of these

Correct answer: None of these

Solution

Let Z = x + iy, A = 3+4i, B = -2+7i, C = 5-2i. The function f(Z) = |Z-A|² + |Z-B|² + |Z-C|² = 3|Z|² - 2*Re(Z*(Abar+Bbar+Cbar)) + |A|²+|B|²+|C|², minimized when Z = (A+B+C)/3. A+B+C = (3-2+5) + (4+7-2)i = 6 + 9i. Z_min = (6+9i)/3 = 2+3i. None of the options (1+3i, 3+3i, 3+4i) equals 2+3i. Answer: None of these.

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