$\frac{8}{\sqrt{13}}$

$\frac{4}{\sqrt{13}}$

$\frac{\sqrt{17}}{8}$

$\frac{8}{\sqrt{17}}$

$5 x^2+5 y^2-8 x-14 y-32=0$

$x^2+y^2-12 x-14 y-28=0$

$3 x^2+3 y^2-8 x-14 y-31=0$

$x^2+y^2-5 x-7 y-14=0$

There will be 3 common tangents to $C_1$ to $C_2$

There will be exactly two common tangents to $C_1$ and $C_2$

There will be no common tangent or there will be exactly two common tangents to $C_1$ and $C_2$

There will be no common tangents or there will be four common tangents to $C_1$ and $C_2$

a circle with centre at $\left(-1, \frac{1}{2}\right)$

a pair of lines intersecting at $(-1,1)$

a parabola with focus at $(-1,1)$

a hyperbola with centre at $(-1,1)$