Answer
(a) Vertices: $V(0,±1)$
Foci: $F(0,±\frac{\sqrt 2}{2})$
Eccentricity:
$e=\frac{\sqrt 2}{2}$
(b)
Length of the major axis:
$2a=2$
Length of the minor axis:
$2b=\sqrt 2$
(c)
Work Step by Step
$y^2=1-2x^2$
$2x^2+y^2=1$
$\frac{x^2}{\frac{1}{2}}+\frac{y^2}{1}=1$
$\frac{x^2}{(\frac{\sqrt 2}{2})^2}+\frac{y^2}{1^2}=1$
The major axis is vertical.
Equation of an ellipse when major axis is vertical (center at the origin):
$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$
So:
$a=1$
$b=\frac{\sqrt 2}{2}$
$c^2=a^2-b^2=1-\frac{1}{2}=\frac{1}{2}$
$c=\frac{\sqrt 2}{2}$
(a)
Vertices: $V(0,±a)=V(0,±1)$
Foci: $F(0,±c)=F(0,±\frac{\sqrt 2}{2})$
Eccentricity:
$e=\frac{c}{a}=\frac{\frac{\sqrt 2}{2}}{1}=\frac{\sqrt 2}{2}$
(b)
Length of the major axis:
$2a=2$
Length of the minor axis:
$2b=\sqrt 2$
