For $z \in \mathbb{C}$,if the minimum value of $(|z-3 \sqrt{2}| + |z-p \sqrt{2} i|)$ is $5 \sqrt{2}$,then a value of $p$ is $.......$

$|{z_1} + {z_2}| = |{z_1}| + |{z_2}|$ is possible if

$A$ circle whose radius is $r$ and centre is $z_0$,then the equation of the circle is

Let $C$ denote the set of all complex numbers. Define $A = \{(z, w) \mid z, w \in C \text{ and } |z| = |w|\}$ and $B = \{(z, w) \mid z, w \in C \text{ and } z^2 = w^2\}$. Then:

$\alpha$ is the real root and $\beta, \gamma$ are the other roots of the equation $x^3-a^3=0$ $(a>0)$. Then the number of common points of the curves given by $|z-\beta|=\frac{\sqrt{3} a}{2}$ and $|z-\gamma|=\frac{\sqrt{3} a}{2}$ is

If $\log_{\sqrt{3}} \left( \frac{|z|^2 - |z| + 1}{2 + |z|} \right) < 2$,then the locus of $z$ is