Answer
(a)
Center: $C(1,-2)$
$V_1(1,-2-2\sqrt 2)$ and $V_2(1,-2+2\sqrt 2)$
$F_1(1,-4)$ and $F_2(1,0)$
(b)
Length of the major axis:
$2a=4\sqrt 2$
Length of the minor axis:
$2b=4$
(c)
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is vertical):
$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$
$2x^2+y^2=2+4(x-y)$
$2x^2-4x+y^2+4y=2$
$2x^2-4x+2+y^2+4y+4=2+2+4$
$2(x-1)^2+(y+2)^2=8$
$\frac{(x-1)^2}{4}+\frac{(y+2)^2}{8}=1$
$\frac{(x-1)^2}{2^2}+\frac{[y-(-2)]^2}{(2\sqrt 2)^2}=1$
$h=1$
$k=-2$
Center: $C(1,-2)$
$a=2\sqrt 2$
$b=2$
$c^2=a^2-b^2=8-4=4$
$c=2$
The given equation can be obtained by shifting
$\frac{x^2}{2^2}+\frac{y^2}{(2\sqrt 2)^2}=1$
right 1 unit and downward 2 units. In this equation:
Vertices: $V(0,±a)$
$V_1(0,-2\sqrt 2)$ and $V_2(0,2\sqrt 2)$
Foci: $F(0,±c)$
$F_1(0,-2)$ and $F_2(0,2)$
Now, shift these points 1 unit to the right and 2 units downward:
$V_1(1,-2-2\sqrt 2)$ and $V_2(1,-2+2\sqrt 2)$
$F_1(1,-4)$ and $F_2(1,0)$
Length of the major axis:
$2a=4\sqrt 2$
Length of the minor axis:
$2b=4$
