Answer
Center: $(-3,1)$
Foci: $(-3-2\sqrt 2,1),(-3+2\sqrt 2,1)$
Vertices: $(-6,1),(0,1)$
See graph
Work Step by Step
We are given the ellipse:
$x^2+9y^2+6x-18y+9=0$
Put the equation in standard form:
$(x^2+6x+9)-9+9(y^2-2y+1)-9+9=0$
$(x+3)^2+9(y-1)^2=9$
$\dfrac{(x+3)^2}{9}+\dfrac{9(y-1)^2}{9}=1$
$\dfrac{(x+3)^2}{9}+\dfrac{(y-1)^2}{1}=1$
The equation is in the form:
$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$
Identify $h,k,a,b$:
$h=-3$
$k=1$
$a^2=9\Rightarrow a=\sqrt{9}=3$
$b^2=1\Rightarrow b=1$
Determine $c$:
$a^2=b^2+c^2$
$c^2=a^2-b^2$
$c^2=9-1$
$c^2=8$
$c=\sqrt 8=2\sqrt 2$
Determine the center:
$(h,k)=(-3,1)$
Determine the foci:
$(h-c,k)=(-3-2\sqrt 2,1)$
$(h+c,k)=(-3+2\sqrt 2,1)$
Determine the vertices:
$(h-a,k)=(-3-3,1)=(-6,1)$
$(h+a,k)=(-3+3,1)=(0,1)$
Determine the $y$-intercepts:
$(h,k-b)=(-3,1-1)=(-3,0)$
$(h,k+b)=(-3,1+1)=(-3,2)$
Graph the ellipse:
