Answer:
[tex]\dfrac{x^2}{8}+\dfrac{y^2}{4}=1[/tex]
Step-by-step explanation:
The relationship between the focus, directrix, and semi-major axis is ...
foci are ±ae
directrices are ±a/e
where "a" is the length of the semi-major axis and e is the eccentricity.
Using the given focus and directrix locations, we have ...
ae = 2
a/e = 4
When we multiply these equations, we have ...
(ae)(a/e) = (2)(4)
a^2 = 8
This is the denominator of the x^2 term in the equation for an ellipse:
x^2/a^2 + y^2/b^2 = 1
_____
We can find b^2 from ...
ae = √(a^2 -b^2)
b^2 = (a^2 -(ae)^2) = 8 -2^2 = 4
So, our ellipse equation is ...
x^2/8 +y^2/4 = 1 . . . . . . matches the first choice
