A point z moves in the complex plane such that arg((z - 2)/(z + 2))...

Given that

$arg\left(\dfrac{z - 2}{z + 2}\right) = \dfrac{\pi}{4}$

Let $z = x + iy$
$⇒ arg\left(\dfrac{x + iy - 2}{x + iy + 2}\right) = \dfrac{\pi}{4}$

$⇒ arg\left(\left(\dfrac{x + iy - 2}{x + iy + 2}\right)\left(\dfrac{x + 2 - iy}{x + 2 - iy}\right)\right) = \dfrac{\pi}{4}$

$⇒ arg\left(\dfrac{{x}^{2} + {y}^{2} - 4 + 4iy}{( x + 2{)}^{2} + {y}^{2}}\right) = \dfrac{\pi}{4}$

$⇒ \dfrac{{x}^{2} + {y}^{2} - 4 + 4iy}{( x + 2{)}^{2} + {y}^{2}} = \tan \left(\dfrac{\pi}{4}\right)$

$⇒ \dfrac{{x}^{2} + {y}^{2} - 4}{4y} = 1$

${x}^{2} + ( y - 2{)}^{2} = 8$ ……….(1)

Equation $( 1 )$ represents circle, whose centre $( 0, 2 )$, radius $2\sqrt{2}$.

Minimum distance $d = | AC - r |$
Given point $A ( 9\sqrt{2}, 2 )$

$= | AC - r | = ∣ ( 9\sqrt{2} - 2\sqrt{2} ) = 7\sqrt{2}$

$⇒$ Square of distance $= {d}^{2} = ( 7\sqrt{2}{)}^{2} = 98$