Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ - and $z$-axes, respectively, is half of the angle that this line makes with the positive $x$-axes. Then the sum of all possible values of the angle $\beta$ is

$$If\,\,{z_1},{z_2},{z_3} \in \,\,are\,\,the\,\,vertices\,\,of\,\,an\,\,equilateral\,\,triangle,\,\,whose\,\,centroid\,\,is\,\,{z_0},\,\,then\,\,\sum\limits_{k = 1}^3 {{{\left( {{z_k} - {z_0}} \right)}^2}\,is\,\,equal\,\,to} $$

Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $( \pm 2,0)$. Let $P Q R$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $Q R$ of length $2 a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $P Q R$ is :

The shortest distance between the curves $y^2=8 x$ and $x^2+y^2+12 y+35=0$ is: