Complex complication

Algebra Level 5

\( z_1\) and \(z_2\) are two constant complex numbers such that \(\displaystyle | z_1 - z_2 | = 5 \). A complex number, \(z\) , satisfies the following equation.

\(\displaystyle | z + \sqrt{ z^2 + z_1 z_2 - zz_1 - zz_2 } - \frac{z_1 + z_2}{2} | + | z - \sqrt{ z^2 + z_1 z_2 - zz_1 - zz_2 } - \frac{z_1 + z_2}{2} | = 10 \)

The locus of another complex number \(z_3\) is given as :
For the equation \( \displaystyle \tan ( \text{ arg} ( z - z_3 ) ) = \tan \phi \), there exists exactly one value of \(z\) for two different values of \(\phi \), \( ( \phi _1 \) and \( \phi _2 ) \) such that \( \displaystyle | \phi _1 - \phi _2 | = m\pi + \frac{\pi }{2} \quad ; m \in \mathbb{Z}\).
( The value of the complex number \(z\) is not same for the two values of \(\phi \) but the number of solutions is unity for the two values. )

The value \(\displaystyle |2z_3 - z_1 -z_2 | \) is found to be constant. Enter the answer as the square of the constant value.

Details and Assumptions:

  • \(\text{arg }(z)\) gives the principal argument of a complex number \(z\).
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