Find the distance between the foci of the following hyperbola.
\[
H\colon \frac{\left (x + 1\right )^{2}}
{16} -\frac{\left (y + 5\right )^{2}}
{9} = 1
\]
Find the distance between the foci of the following hyperbola.
\[
H\colon \frac{\left (x + 3\right )^{2}}
{9} -\frac{\left (y - 2\right )^{2}}
{27} = 1
\]
Find the slope of a line through the center of the hyperbola
\[
\frac{(x - 2)^{2}}
{4} -\frac{(y + 3)^{2}}
{9} = 1
\]
which has a unique intersection with this hyperbola.
There is no solution, the line with these properties does not exist.
Among the following lines (which all pass through the point
\([-1;3]\))
identify a line which is tangent to the following hyperbola.
\[
(x + 2)\cdot (y - 2) = 1
\]
Parabola is a set of the points that are equidistant from the point (focus)
and the line (directrix). Find the equation of the directrix of the parabola
\(P\colon x^{2} - 4x - 6y - 17 = 0\).