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Question

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 |z1z2|2, where z1,z2S with Re(z1)>0 and Re(z2)<0, is

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Solution

z4|z|4=4iz2
z4z2 ¯¯¯z2=4iz2 (z¯¯¯z=|z|2)
z2(z2(¯¯¯z)24i)=0
z2=0 or z2(¯¯¯z)2=4i
Let z=x+iy
z2(¯¯¯z)2=4i
(x+iy)2(xiy)2=4i
x2y2+2ixy(x2y22ixy)=4i
4ixy=4i
xy=1

Now, |z1z2|2=(x1x2)2+(y1y2)2
=x21+x22+y21+y222x1x22y1y2
=x21+x22+y21+y22+2x1(x2)+2y1(y2)
Applying A.M.G.M.,
(x21+x22+y21+y22+x1(x2)+x1(x2)+y1(y2)+y1(y2))8
(x21 x22 y21 y22 x21 x22 y21 y22)1/8
[(x1y1)(x2 y2)]1/2=1

|z1z2|28


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