If the circles $x^2+y^2-2 x-2(3+\sqrt{7}) y+8+6 \sqrt{7}=0$ and $x^2+y^2-8 x-6 y+k^2=0, k \in \mathbf{Z}$, have exactly two common tangents, then the number of possible values of $k$ is

The circle $S=0$ cuts the circles

$C_1=x^2+y^2-8 x-2 y+16=0$ and $C_2=x^2+y^2-4 x-4 y-1=0$ orthogonally. If the common chord of $S=0$ and $C_1=0$ is $2 x+13 y-15=0$, then the centre of $S=0$ is

The equation of the circle passing through the points of intersection of the two orthogonal circles $S_1=x^2+y^2+k x-4 y-1=0$, $S_2=3 x^2+3 y^2-14 x+23 y-15=0$ and passing through the point $(-1,-1)$ is

Consider the parabola $y^2+2 x+2 y-3=0$ and match the items of List-I with those of the List-II.

$$ \begin{array}{llll} \hline & \text { List-I } & & \text { List-II } \\ \hline \text { A. } & 2 x-5=0 & \text { I. } & \text { Vertex } \\ \hline \text { B. } & \left(\frac{3}{2},-1\right) & \text { II. } & \text { Focus } \\ \hline \text { C. } & y+1=0 & \text { III. } & \text { Equation of directrix } \\ \hline \text { D. } & (2,-1) & \text { IV. } & \text { Equation of the axis } \\ \hline & & \text { V. } & \text { Equation of the Latus rectum } \\ \hline \end{array} $$