Which of the following is correct?

The area of $\Delta$ whose vertices are $z, \omega z, z + \omega z$ is (where $\omega$ is a complex cube root of unity):

If the amplitude of $z-2-3i$ is $\pi/4$,then the locus of $z=x+iy$ is

If $S = \{z \in \mathbb{C} : |z - i| = |z + i| = |z - 1|\}$,then $n(S)$ is:

Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1| = 9$ and $|z_2 - (3 + 4i)| = 4$. Then the minimum value of $|z_1 - z_2|$ is

$A$ function $f$ is defined on the complex numbers by $f(z) = (a + ib)z$,where $a, b \in \mathbb{R}^+$. This function has the property that the $f$-image of any point in the complex plane is equidistant from that point and the origin. If $|a + bi| = 10$ and $b^2 = \frac{p}{q}$,where $p, q \in \mathbb{Z}$ and $\text{gcd}(p, q) = 1$,then $p + q$ is: