## Task B-2.1.

Among all complex numbers $z$ that satisfy the equation

$$
\left|\frac{z-i}{z-3 i}\right|=\frac{1}{2}
$$

determine the one with the largest modulus.
First solution.

Let $z=x+y i$. From the given condition, we have

$$
\begin{gathered}
\left|\frac{x+(y-1) i}{x+(y-3) i}\right|=\frac{1}{2} \\
\sqrt{x^{2}+(y-1)^{2}}=\frac{1}{2} \sqrt{x^{2}+(y-3)^{2}}
\end{gathered}
$$

After squaring and performing the necessary calculations, we find that for the complex number $z$,

$$
x^{2}+y^{2}-\frac{2}{3} y-\frac{5}{3}=0
$$

which can be written in the form

$$
x^{2}+\left(y-\frac{1}{3}\right)^{2}=\left(\frac{4}{3}\right)^{2}
$$

We conclude that $z$ lies on a circle with center on the $y$-axis at $S\left(0, \frac{1}{3}\right)$, with radius $r=\frac{4}{3}$. The point on this circle that is farthest from the origin is the desired point. Therefore, the solution is a complex number whose real part is 0 and whose imaginary part is $\frac{5}{3}$, i.e., the number $z=\frac{5}{3} i$.