Find the distance between the vertices of the following hyperbola.
\[
H\colon \frac{\left (x + 1\right )^{2}}
{25} -\frac{\left (y + 2\right )^{2}}
{16} = 1
\]
Find the distance between the vertices of the following hyperbola.
\[
H\colon \frac{\left (x - 3\right )^{2}}
{16} -\frac{\left (y + 2\right )^{2}}
{25} = 1
\]
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.
\[
H\colon \frac{\left (x + 3\right )^{2}}
{9} -\frac{\left (y - 2\right )^{2}}
{27} = 1
\]
Given points \(A = [0;1;2]\),
\(B = [1;2;0]\),
\(C = [1;2;3]\), find the angle
between the lines \(AB\)
and \(AC\).
Round your answer to the nearest degree.
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
\(P\colon x^{2} - 4x - 6y - 17 = 0\).
Find the angle between the \(x\)-axis
and the line \(p\).
\[
\begin{aligned}p\colon x& = 2 - t, &
\\y & = 3t,
\\z & = 1;\ t\in \mathbb{R}
\\ \end{aligned}
\]
Round your answer to the nearest minute.
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
\(P\colon y^{2} + 4y + 4x - 4 = 0\).