In the Argand plane, the points z with |z - 2i| = 2 |z + 1| lie on a circle…

Square both sides with $z = x + iy$: $x^2 + (y - 2)^2 = 4\left[(x + 1)^2 + y^2\right]$, which becomes $3 X^{2} + 8 X + 3 Y^{2} + 4 Y = 0$, or, dividing by $3$, $X^{2} + \frac{8 X}{3} + Y^{2} + \frac{4 Y}{3} = 0$ (with $x$ and $y$ written as $X$ and $Y$).

(a), (b), (c) The centre is $\left(- \frac{4}{3}, - \frac{2}{3}\right)$, that is $(-1.33, -0.67)$, and the radius is $\sqrt{\frac{20}{9}} = \frac{2 \sqrt{5}}{3} = 1.49$.

(d) The equation has no constant term, so the circle passes through the origin. The farthest point from the origin is then at the other end of the diameter through it: $|z|_{\max} = 2r = \frac{4 \sqrt{5}}{3} = 2.98$.

(e) The rightmost point is the centre moved by one radius: $- \frac{4}{3} + \frac{2 \sqrt{5}}{3} = 0.16$.