For a complex number z, let Re(z) denote the real part of z. Let S be the set of all complex numbers z satisfying z4−|z|4=4iz2, where i=√−1. Then the minimum possible value of |z1−z2|2, where z1,z2∈S with Re(z1)>0 and Re(z2)<0, is
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Solution
z4−|z|4=4iz2 ⇒z4−z2¯¯¯z2=4iz2(∵z¯¯¯z=|z|2) ⇒z2(z2−(¯¯¯z)2−4i)=0 ⇒z2=0 or z2−(¯¯¯z)2=4i
Let z=x+iy z2−(¯¯¯z)2=4i ⇒(x+iy)2−(x−iy)2=4i ⇒x2−y2+2ixy−(x2−y2−2ixy)=4i ⇒4ixy=4i ⇒xy=1