Question

determine the vertices, asymptotes and foci of the hyperbola: 9(x+2)^(2)/25-(y+2)^(2)/9=1

Answer 1

\[
\frac{9(x+2)^{2}}{25}-\frac{(y+2)^2}{9}=1\\
\frac{(x+2)^{2}}{\frac{25}{9}}-\frac{(y+2)^2}{9}=1\\
a=\frac 53\\
b= 3\\
c^2 = \frac {25}9 + 9\\
c^2= \frac{25+81}9\\
c=\frac {\sqrt{106}}3
\]
What are the foci?

Answer 2

The vertices are on y=-2.
\[
\left( -2\pm \frac 5 3, -2 \right)
\]

Answer 3

The foci are
\[
\left(-2\pm\frac{\sqrt{106}}{3},-2 \right)
\]

Answer 4

The red point is the center.
The Blue points are the vertices.
The green points are the foci.
The lines are the two asymptotes
\[
y=\pm \frac 9 5( x+2) -2\\
\]