# 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 $$z$$.
×