For all complex numbers $z_1$ and $z_2$ satisfying $|z_1| = 12$ and $|z_2 - (3 + 4i)| = 5$,the minimum value of $|z_1 - z_2|$ is:

If a point $P$ denotes a complex number $z=x+iy$ in the Argand plane and if $\frac{z+1}{z+i}$ is a purely real number,then the locus of $P$ is

For any two non-zero complex numbers $z_1$ and $z_2$, if $|z_1+z_2|^2=|z_1|^2+|z_2|^2$, then

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:

If $S = \{z \in \mathbb{C} : \frac{z-i}{z+2i} \in \mathbb{R}\}$,then:

If $z_1$ and $z_2$ are two roots of the equation $z^2+az+b=0$ with $a^2 < 4b$, then the origin, $z_1$, and $z_2$ form an equilateral triangle if