Answer
$ a.\qquad$
New center: $\quad (-3, -2)$
New foci: $\quad (-3,-6)$ and $(-3,2)$
New vertices: $\quad (-3,-7)$ and $(-3,3)$
$ b.\qquad$
See graph.
Work Step by Step
$ a.\qquad$
$\displaystyle \frac{x^{2}}{9}+\frac{y^{2}}{25}=1$
Which is of the form $\displaystyle \quad\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1 \quad (a\gt b)$
Foci on the x-axis,
Center: $(0,0),\quad a=5,\quad b=3$
Center-to-focus distance: $\quad c=\sqrt{a^{2}-b^{2}}=\sqrt{25-9}=4$
Foci: $\quad(0,\pm c)= \quad(0,\pm 4)$
Vertices: $\quad (0,\pm a)= \quad (0,\pm 5)$
The translations are such that $(x,y)\rightarrow(x',y')$, where
$\left\{\begin{array}{ll}
x'=x-3 & \text{... shift left}\\
y'=y-2 & \text{... shift down}
\end{array}\right.$
New center: $(-3+0,-2+0)=(-3,-2)$
New foci: $\quad(-3+0-2\pm c)= \quad(-3,-2\pm 4)$,
$(-3,-6)$ and $(-3,2)$
New vertices: $\quad (-3+0,-2\pm a)= \quad (-3,-2\pm 5)$
$(-3,-7)$ and $(-3,3)$
$ b.\qquad$
See graph.
