For a non –zero complex number Z let Re(Z) denotes the real part of z ,let S be the set of all complex numbers Z satisfying $Z^{4} - {| Z |}^{4} = 12 i {(Z)}^{2} w h e r e i = \sqrt{- 1} . i f Z_{1}, Z_{2} \in S a n d Re (Z_{1}) > 0, Re (Z_{2}) < 0$
then which of the following options is /are correct ?

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Minimum value of $| Z_{1} - Z_{2} | i s \sqrt{6}$

Minimum value of $\sqrt{| Z_{1} - Z_{2} |} i s 2 \sqrt{6}$

Area of triangle formed by any tangent of locus of Z with coordinate axes is 6

Area of triangle formed by any tangent of locus of Z with coordinate axes is 12

answer is B, C.

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Detailed Solution

$Z_{1}$ lies of xy =3 in first quadrant and $Z_{2}$ lies on
xy=3 $Z_{2} \equiv (- \sqrt{3}, \sqrt{3})$
$\Rightarrow | Z_{1} - Z_{2} |$ minimum $2 \sqrt{6}$
Required area of triangle =6

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