The complex number $z = x + iy$ which satisfies the equation $\left| \frac{z - 5i}{z + 5i} \right| = 1$ lies on

If $z$ is any complex number satisfying $|z - 3 - 2i| \leq 2$,then the minimum value of $|2z - 6 + 5i|$ is

Let $P=\{z \in C:|z+2-3 i| \leq 1\}$ and $Q=\{z \in C: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $P \cap Q, |z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2+2|z_2|^2=\alpha+\beta \sqrt{2}$,where $\alpha, \beta$ are integers,then $\alpha+\beta$ equals . . . . . . .

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

The roots of the cubic equation $(z + ab)^3 = a^3$,where $a \neq 0$,represent the vertices of a triangle of sides of length:

The locus of the point representing the complex number $z$ for which $|z+3|^2-|z-3|^2=15$ is