Answer:
The equation of the ellipse is:
(x^2)/16 + (y^2)/4 = 1
Explanation:
An ellipse is a set of points on a plane, the sum of whose distances from two fixed points (called foci) is constant. The distance between the foci of an ellipse is denoted by 2c, and the distance between the center of the ellipse and one of its vertices is denoted by a.
In this problem, the foci are located at (2,0) and (-2,0), so the distance between the foci is 2c = 4, which means that c = 2. The major vertices of the ellipse are located at (4,0) and (-4,0), so the distance between the center and one of the vertices is a = 4.
The formula for the equation of an ellipse centered at the origin is:
(x^2)/(a^2) + (y^2)/(b^2) = 1
where a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex. Since the center of this ellipse is at the origin and the major axis lies on the x-axis, we know that b = a, so we can substitute a for b in the equation:
(x^2)/(a^2) + (y^2)/(a^2) = 1
To find a, we use the fact that c^2 = a^2 - b^2:
a^2 - b^2 = c^2
a^2 - a^2 = 4
b^2 = 4
b = 2
Now we can substitute a = 4 and b = 2 into the equation:
(x^2)/(16) + (y^2)/(4) = 1
This is the equation of the ellipse.