Interpret the following equations geometrically on the Argand plane.

$1 < |z - 2 - 3 i| < 4$

If $a, b, c, d$ be a form consecutive term of an increasing A.P., then the roots of the equation $\left( {x - a} \right)\left( {x - c} \right) + 2\left( {x - b} \right)\left( {x - d} \right) = 0$

If $z = x+iy$ and $w = \dfrac{(1-iz)}{(z-i)}$, then $|w| = 1$ implies that, in the complex plane

If $z_1+ z_2 + z_3 = 0$ and $|z_1|=|z_2|=|z_3|= 1$, then area of triangle whose vertices are $z_1, z_2$ and $z_3$ is:

If $\alpha$ and $\beta$ are two different complex numbers with $|\beta|=1$, then $\left | \dfrac{\beta -\alpha}{1-\bar{\alpha }\beta } \right |$ is equal to.