Respuesta :
Answer:
[tex]\frac{x^2}{8} + \frac{y^2}{4}=1[/tex]
Step-by-step explanation:
The directrices in this case are vertical lines, so we have a horizontal ellipse. The equation for that ellipse is:
[tex]\frac{(x - h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1[/tex]
The center of the ellipse is (h,k), the diretrix is x = d and the foci are given by (h+c, k) and (h-c, k)
So, comparing the foci, we have that k = 0 and:
[tex]h + c = 2[/tex]
[tex]h - c = -2[/tex]
Adding these two equations, we have:
[tex]2h = 0[/tex]
[tex]h = 0[/tex]
[tex]c = 2[/tex]
We can find the value of a^2 using the property:
[tex]c / a = a / d[/tex]
Using c = 2 and d = 4, we have:
[tex]a^2 = c * d[/tex]
[tex]a^2 = 8[/tex]
Now, to find b^2, we use the property:
[tex]a^2 = b^2 + c^2[/tex]
[tex]8 = b^2 + 4[/tex]
[tex]b^2 = 4[/tex]
So the equation of the ellipse is:
[tex]\frac{x^2}{8} + \frac{y^2}{4}=1[/tex]
