Answer
$(0,-2), (0,2)$
Work Step by Step
We are given the curves:
$\begin{cases}
4x^2+y^2=4\\
y^2-4x^2=4
\end{cases}$
The equation $4x^2+y^2=4$ represents an ellipse centered in origin. The equation $y^2-4x^2=4$ represents a vertical hyperbola centered in origin.
We graph both curves.
From the graph we find the intersection points:
$(0,-2)$ and $(0,2)$
We check if both points check the equations:
$(0,-2)$
$4x^2+y^2=4$
$4(0^2)+(-2)^2\stackrel{?}{=}4$
$4=4\checkmark$
$y^2-4x^2=4$
$(-2)^2-4(0^2)\stackrel{?}{=}4$
$4=4\checkmark$
$(0,2)$
$4x^2+y^2=4$
$4(0^2)+2^2\stackrel{?}{=}4$
$4=4\checkmark$
$y^2-4x^2=4$
$2^2-4(0^2)\stackrel{?}{=}4$
$4=4\checkmark$
So the system's solutions are:
$(0,-2), (0,2)$
