If ${z_1} = 10 + 6i$,${z_2} = 4 + 6i$ and $z$ is a complex number such that $\text{amp}\left( \frac{z - z_1}{z - z_2} \right) = \frac{\pi}{4}$,then the value of $|z - 7 - 9i|$ is equal to

$z_1$ and $z_2$ are the roots of $3z^2 + 3z + b = 0$. If the origin,$A(z_1)$,and $B(z_2)$ form an equilateral triangle,then the value of $b$ is equal to

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:

Let $\arg(z)$ represent the principal argument of the complex number $z$. The curves $|z|=3$ and $\arg(z-1)-\arg(z+1)=\frac{\pi}{4}$ intersect:

If the vertices of a quadrilateral are $A = 1 + 2i,$ $B = -3 + i,$ $C = -2 - 3i,$ and $D = 2 - 2i,$ then the quadrilateral is:

If the roots of the equation $Z^3+i Z^2+2 i=0$ are the vertices of a triangle $ABC$,then that triangle $ABC$ is