Equation of a common tangent to the circle $x^2+y^2=4$ and to the ellipse $2 x^2+25 y^2=50$ is

The $\theta$ is the angle made by the common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with positive $X$-axis, then $\cos 2 \theta=$

For the hyperbola $x^2-y^2-4 x+2 y+c=0$, if the focus is $S(2+2 \sqrt{2}, k)$ and the directrix that is adjacent to $S$ is $x=2+\sqrt{2}$, then $c=$

The quadrilateral formed by the points $A(1,2,5), B(-1,6,1), C(3,4,-3)$ and $D(5,0,1)$ is a