Let $u = \frac{2z + i}{z - ki}$, where $z = x + iy$ and $k > 0$. If the curve represented by $\operatorname{Re}(u) + \operatorname{Im}(u) = 1$ intersects the $y$-axis at the points $P$ and $Q$ such that $PQ = 5$, then the value of $k$ is:

Convert the given complex number into polar form: $-3$.

If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1$,then the origin and the points represented by $z_{1}$ and $z_{2}$:

If $\left|\frac{z}{1+i}\right|=2$,where $z=x+iy$ and $i=\sqrt{-1}$ represents a circle,then the centre $C$ and radius $r$ of the circle are:

In the Argand plane,the values of $z$ satisfying the equation $|z-1|=|i(z+1)|$ lie on

If ${z_1}, {z_2}, {z_3}$ are three non-zero complex numbers such that ${z_2} \neq {z_1}$,$a = |{z_1}|$,$b = |{z_2}|$,and $c = |{z_3}|$. Suppose that $\left| {\begin{array}{*{20}{c}} a & b & c \\ b & c & a \\ c & a & b \end{array}} \right| = 0$,then $arg\left( {\frac{{{z_3}}}{{{z_2}}}} \right)$ is equal to: