A point $\mathrm{z}$ moves in the complex plane such that $\arg \left(\frac{\mathrm{z}-2}{\mathrm{z}+2}\right)=\frac{\pi}{4}$, then the minimum value of $|\mathrm{z}-9 \sqrt{2}-2 i|^{2}$ is equal to______.
Let $z=x+i y$
$\arg \left(\frac{x-2+i y}{x+2+i y}\right)=\frac{\pi}{4}$
$\arg (x-2+i y)-\arg (x+2+i y)=\frac{\pi}{4}$
$\tan ^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}-2}\right)-\tan ^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}+2}\right)=\frac{\pi}{4}$
$\frac{\frac{\mathrm{y}}{\mathrm{x}-2}-\frac{\mathrm{y}}{\mathrm{x}+2}}{1+\left(\frac{\mathrm{y}}{\mathrm{x}-2}\right) \cdot\left(\frac{\mathrm{y}}{\mathrm{x}+2}\right)}=\tan \frac{\pi}{4}=1$
$\frac{x y+2 y-x y+2 y}{x^{2}-4+y^{2}}=1$
$4 y=x^{2}-4+y^{2}$
$x^{2}+y^{2}-4 y-4=0$
locus is a circle with center $(0,2) \&$ radius $=2 \sqrt{2}$
min. value $=(\mathrm{AP})^{2}=(\mathrm{OP}-\mathrm{OA})^{2}$
$=(9 \sqrt{2}-2 \sqrt{2})^{2}$
$=(7 \sqrt{2})^{2}=98$
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