Let z be a complex number such that |(z - 2i)/(z + i)| = 2, z ≠ - i...

Given equation is

$\left|\dfrac{z - 2i}{z + i}\right| = 2, z \ne - i$

On simplifying the equation, we get

$( z - 2i ) ( \overset{¯}{z} + 2i ) = 4 ( z + i ) ( \overset{¯}{z} - i )$

$⇒ z\overset{¯}{z} + 4 + 2i ( z - \overset{¯}{z} ) = 4 ( z\overset{¯}{z} + 1 + i ( \overset{¯}{z} - z ) )$

$⇒ 3z\overset{¯}{z} - 6i ( z - \overset{¯}{z} ) = 0$

Now putting the value of $z = x + iy \text{ and } \overset{¯}{z} = x - iy$ we get,

$⇒ {x}^{2} + {y}^{2} - 2i ( 2iy ) = 0$

$⇒ {x}^{2} + {y}^{2} + 4y = 0$

On comparing the above equation with the general equation of the circle i.e

$( x - h{)}^{2} + ( y - k{)}^{2} = {r}^{2}$

We get, centre $( 0, - 2 )$