Answer:
a) [tex]\frac{x^2}{16} +\frac{ y^2}{12} = 1[/tex]
b) [tex]\frac{x^2}{64} +\frac{ y^2}{16} = 1[/tex]
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:
[tex]\frac{(x - h)^2}{a^2} +\frac{ (y - k)^2}{b^2} = 1[/tex]
The coordinates of the foci and the vertices are given by:
Foci: [tex]F(h \pm c, k)[/tex]
Vertices: [tex]V(h\pm a, k)[/tex]
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:
[tex]c^2 = a^2 - b^2[/tex]
[tex]4 = 16 - b^2[/tex]
[tex]b^2 =12[/tex]
So the equation of the ellipse is:
[tex]\frac{x^2}{16} +\frac{ y^2}{12} = 1[/tex]
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:
[tex]\frac{x^2}{64} +\frac{ y^2}{16} = 1[/tex]
