QJEEAdv20Maae3a: JEE Advanced 2020 Mathematics - Class 11 JEEAdv20Maae3a Solution: For a complex number z, let Re(z) denote that real part of z.…

For a complex number z, it is given that,

${z^4} - |z{|^4} = 4i{z^2}$

$\Rightarrow {z^4} - {z^{2 - 2}}z = 4i{z^2}$

$\Rightarrow {z^2}(z - \overline z )(z + \overline z ) = 4i{z^2}$

So, either ${z^2} = 0$ or $(z - \overline z )(z + \overline z ) = 4i$

Now, Case - I, if ${z^2} = 0$ and $z = x + iy$

So, ${x^2} - {y^2} + 2ixy = 0$

$\Rightarrow {x^2} - {y^2} = 0$

and $xy = 0$

$\Rightarrow x = y = 0$

$\Rightarrow z = 0$ , which is not possible according to given conditions.

Case - II, if $(z - \overline z )(z + \overline z ) = 4i$ and

$z = x + iy$

So, $(2iy)(2x) = 4i$

$\Rightarrow$ xy = 1 is an equation of rectangular hyperbola and for minimum value of $|{z_1} - {z_2}{|^2}$ , the z₁ and z₂ must be vertices of the rectangular hyperbola.

Therefore, ${z_1} = 1 + i$ and ${z_2} = - 1 - i$

$\therefore$ Minimum value of $|{z_1} - {z_2}{|^2}$

$= {(1 + 1)^2} + {(1 + 1)^2} = 4 + 4 = 8$ .