Answer
(a)
Center: $C(0,-5)$
Vertices: $V_1(0,-10)$ and $V_2(0,0)$
Foci: $F_1(0,-9)$ and $F_2(0,-1)$
(b)
Length of the major axis:
$2a=10$
Length of the minor axis:
$2b=6$
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is vertical):
$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$
$\frac{x^2}{9}+\frac{(y+5)^2}{25}=1$
$\frac{(x-0)^2}{3^2}+\frac{[y-(-5)]^2}{5^2}=1$
$h=0$
$k=-5$
Center: $C(0,-5)$
$a=5$
$b=3$
$c^2=a^2-b^2=25-9=16$
$c=4$
The given equation can be obtained by shifting
$\frac{x^2}{3^2}+\frac{y^2}{5^2}=1$
downward 5 units. In this equation:
Vertices: $V(0,±a)$
$V_1(0,-5)$ and $V_2(0,5)$
Foci: $F(0,±c)$
$F_1(0,-4)$ and $F_2(0,4)$
Now, shift these points 5 units downward:
Vertices: $V_1(0,-10)$ and $V_2(0,0)$
Foci: $F_1(0,-9)$ and $F_2(0,-1)$
Length of the major axis:
$2a=10$
Length of the minor axis:
$2b=6$
