Answer
(a)
Center: $C(-1,-1)$
Vertices: $V_1(0,-9)$ and $V_2(0,7)$
Foci: $F_1(0,-1-2\sqrt 7)$ and $F_2(0,-1-2\sqrt 7)$
(b)
Length of the major axis:
$2a=16$
Length of the minor axis:
$2b=12$
(c)
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is vertical):
$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$
$\frac{(x+1)^2}{36}+\frac{(y+1)^2}{64}=1$
$\frac{[x-(-1)]^2}{6^2}+\frac{[y-(-1)]^2}{8^2}=1$
$h=-1$
$k=-1$
Center: $C(-1,-1)$
$a=8$
$b=6$
$c^2=a^2-b^2=64-36=28$
$c=2\sqrt 7$
The given equation can be obtained by shifting
$\frac{x^2}{6^2}+\frac{y^2}{8^2}=1$
left 1 unit and downward 1 unit. In this equation:
Vertices: $V(0,±a)$
$V_1(0,-8)$ and $V_2(0,8)$
Foci: $F(0,±c)$
$F_1(0,-2\sqrt 7)$ and $F_2(0,2\sqrt 7)$
Now, shift these points 1 unit to the left and 1 unit downward:
Vertices: $V_1(0,-9)$ and $V_2(0,7)$
Foci: $F_1(0,-1-2\sqrt 7)$ and $F_2(0,-1-2\sqrt 7)$
Length of the major axis:
$2a=16$
Length of the minor axis:
$2b=12$
