Answer:
a) ![\frac{x^2}{16} +\frac{ y^2}{12} = 1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B16%7D%20%2B%5Cfrac%7B%20y%5E2%7D%7B12%7D%20%3D%201)
b) ![\frac{x^2}{64} +\frac{ y^2}{16} = 1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B64%7D%20%2B%5Cfrac%7B%20y%5E2%7D%7B16%7D%20%3D%201)
Step-by-step explanation:
a)
The vertices are located in the x-axis, so we have a horizontal ellipse.
The equation of an ellipse is given by:
![\frac{(x - h)^2}{a^2} +\frac{ (y - k)^2}{b^2} = 1](https://tex.z-dn.net/?f=%5Cfrac%7B%28x%20-%20h%29%5E2%7D%7Ba%5E2%7D%20%2B%5Cfrac%7B%20%28y%20-%20k%29%5E2%7D%7Bb%5E2%7D%20%3D%201)
The coordinates of the foci and the vertices are given by:
Foci: ![F(h \pm c, k)](https://tex.z-dn.net/?f=F%28h%20%5Cpm%20c%2C%20k%29)
Vertices: ![V(h\pm a, k)](https://tex.z-dn.net/?f=V%28h%5Cpm%20a%2C%20k%29)
Comparing the coordinates with the values given, we have that:
h = 0, k = 0, c = 2, a = 4
To find the value of b we can use the following equation:
![c^2 = a^2 - b^2](https://tex.z-dn.net/?f=c%5E2%20%3D%20a%5E2%20-%20b%5E2)
![4 = 16 - b^2](https://tex.z-dn.net/?f=4%20%3D%2016%20-%20b%5E2)
![b^2 =12](https://tex.z-dn.net/?f=b%5E2%20%3D12)
So the equation of the ellipse is:
![\frac{x^2}{16} +\frac{ y^2}{12} = 1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B16%7D%20%2B%5Cfrac%7B%20y%5E2%7D%7B12%7D%20%3D%201)
b)
If the ellipse is centered at the origin, we have:
h = 0, k = 0
The major axis is 'a' and the other axis is 'b', so we have:
a = 8, b = 4.
So the equation is:
![\frac{x^2}{64} +\frac{ y^2}{16} = 1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B64%7D%20%2B%5Cfrac%7B%20y%5E2%7D%7B16%7D%20%3D%201)