When $\frac{z + i}{z + 2}$ is purely imaginary,the locus described by the point $z$ in the Argand diagram is a

If the area of the triangle formed by the points $z, z + iz$ and $iz$ on the complex plane is $18$,then the value of $|z|$ is

The area of the triangle with vertices $A(z)$,$B(iz)$,and $C(z+iz)$ is

Let $z_1$ and $z_2$ be two distinct complex numbers and let $z = (1-t)z_1 + tz_2$ for some real number $t$ with $0 < t < 1$. If $\operatorname{Arg}(w)$ denotes the principal argument of a non-zero complex number $w$,then which of the following are true?
$(A)$ $|z-z_1| + |z-z_2| = |z_1-z_2|$
$(B)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z-z_2)$
$(C)$ $\left|\begin{array}{cc} z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1 \end{array}\right| = 0$
$(D)$ $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z_2-z_1)$

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

If $z$ is a complex number satisfying $|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=4,$ then $|z|$ cannot be