Let $a, b, c, d$ be real numbers such that $|a-b|=2$,$|b-c|=3$,and $|c-d|=4$. Then,the sum of all possible values of $|a-d|$ is

If $P(x, y)$ represents the complex number $z = x + i y$ in the Argand plane and $\operatorname{Arg} \left( \frac{z - 3 i}{z + 4} \right) = \frac{\pi}{2}$,then the equation of the locus of $P$ is

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

$PQ$ and $PR$ are two infinite rays. $QAR$ is an arc. $A$ point lying in the shaded region,excluding the boundary,satisfies:

Let $u = \frac{2z + i}{z - ki}$, where $z = x + iy$ and $k > 0$. If the curve represented by $\operatorname{Re}(u) + \operatorname{Im}(u) = 1$ intersects the $y$-axis at the points $P$ and $Q$ such that $PQ = 5$, then the value of $k$ is:

The area of the region enclosed by the locus of $z$ given by $\text{Arg}(z + i) - \text{Arg}(z - i) = \frac{2\pi}{3}$ and the imaginary axis is: