Answer
(a) center $C(-3, -1)$, vertices $V(-3, -1\pm\sqrt 2)$, foci $F(-3, -1\pm2\sqrt {5})$, asymptotes $y=\pm\frac{1}{3} (x+3)-1$
(b) See graph.
Work Step by Step
(a) Step 1. Rewrite the given equation as $(x^2+6x+9)-9(y^2+2y+1)=-18+9-9$ or $(x+3)^2-9(y+1)^2=-18$ which gives $\frac{(y+1)^2}{2}-\frac{(x+3)^2}{18}=1$
Step 2. Identify the center as $C(-3, -1)$
Step 3. Identify $a=\sqrt 2, b=3\sqrt 2, c=\sqrt {2+18}=2\sqrt {5}$
Step 4. The original vertices are $(0, \pm\sqrt 2)$, vertices after the shift $V(-3, -1\pm\sqrt 2)$
Step 5. Original foci $(0, \pm2\sqrt {5})$, foci after shift $F(-3, -1\pm2\sqrt {5})$
Step 6. Original asymptotes $y=\pm\frac{1}{3} x$, new asymptotes $y=\pm\frac{1}{3} (x+3)-1$
(b) See graph.
