Answer
(a)
Center: $C(2,-3)$
$V_1(2,-9)$ and $V_2(2,3)$
$F_1(2,-3-3\sqrt 3)$ and $F_2(2,-3+3\sqrt 3)$
(b)
Length of the major axis:
$2a=12$
Length of the minor axis:
$2b=6$
(c)
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is vertical):
$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$
$\frac{(x-2)^2}{9}+\frac{(y+3)^2}{36}=1$
$\frac{(x-2)^2}{3^2}+\frac{[y-(-3)]^2}{6^2}=1$
$h=2$
$k=-3$
Center: $C(2,-3)$
$a=6$
$b=3$
$c^2=a^2-b^2=36-9=27$
$c=3\sqrt 3$
The given equation can be obtained by shifting
$\frac{x^2}{3^2}+\frac{y^2}{6^2}=1$
right 2 units and downward 3 units. In this equation:
Vertices: $V(0,±a)$
$V_1(0,-6)$ and $V_2(0,6)$
Foci: $F(0,±c)$
$F_1(0,-3\sqrt 3)$ and $F_2(0,3\sqrt 3)$
Now, shift these points 2 units to the right and 3 units downward:
$V_1(2,-9)$ and $V_2(2,3)$
$F_1(2,-3-3\sqrt 3)$ and $F_2(2,-3+3\sqrt 3)$
Length of the major axis:
$2a=12$
Length of the minor axis:
$2b=6$
