Answer
(a)
Center: $C(0,-2)$
Vertices: $V_1(0,-4)$ and $V_2(0,0)$
Foci: $F_1(0,-2-\sqrt 3)$ and $F_2(0,-2+\sqrt 3)$
(b)
Length of the major axis:
$2a=4$
Length of the minor axis:
$2b=2$
(c)
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is vertical):
$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$
$x^2+\frac{(y+2)^2}{4}=1$
$\frac{(x-0)^2}{1^2}+\frac{[y-(-2)]^2}{2^2}=1$
$h=0$
$k=-2$
Center: $C(0,-2)$
$a=2$
$b=1$
$c^2=a^2-b^2=4-1=3$
$c=\sqrt 3$
The given equation can be obtained by shifting
$\frac{x^2}{1^2}+\frac{y^2}{2^2}=1$
downward 2 units. In this equation:
Vertices: $V(0,±a)$
$V_1(0,-2)$ and $V_2(0,2)$
Foci: $F(0,±c)$
$F_1(0,-\sqrt 3)$ and $F_2(0,\sqrt 3)$
Now, shift these points 2 units downward:
Vertices: $V_1(0,-4)$ and $V_2(0,0)$
Foci: $F_1(0,-2-\sqrt 3)$ and $F_2(0,-2+\sqrt 3)$
Length of the major axis:
$2a=4$
Length of the minor axis:
$2b=2$
