Let $z_1 = 6 + i$ and $z_2 = 4 - 3i$. Let $z$ be a complex number such that $\arg \left( \frac{z - z_1}{z_2 - z} \right) = \frac{\pi}{2}$,then $z$ satisfies -

If $n$ is a positive integer greater than unity and $z$ is a complex number satisfying the equation $z^n = (z + 1)^n$,then

If $z$ is a complex number such that $|z+4| \geq 3$,then the smallest value of $|z+3|$ is

The equation $\text{Re}(z^2) = 1$ represents which of the following?

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + iy$ $(x, y \in \mathbb{R}, i = \sqrt{-1})$ of the equation $sz + t\bar{z} + r = 0$,where $\bar{z} = x - iy$. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ If $L$ has exactly one element,then $|s| \neq |t|$
$(B)$ If $|s| = |t|$,then $L$ has infinitely many elements
$(C)$ The number of elements in $L \cap \{z : |z - 1 + i| = 5\}$ is at most $2$
$(D)$ If $L$ has more than one element,then $L$ has infinitely many elements

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