Answer:
x^2/12 - y^2/4 = 1
Step-by-step explanation:
As the diretrices have simetrical values of x and have y = 0, the center is located at (0,0)
The formula for the diretrices is:
x1 = -a/e and x2 = a/e
And the foci is located at (a*e, 0) and (-a*e, 0)
So we have that:
a/e = 3
a*e = 4
From the first equation, we have a = 3e. Using this in the second equation, we have:
3e*e = 4
e^2 = 4/3
e = 1.1547
Now finding the value of a, we have:
a = 3*1.1547 = 3.4641
Now, as we have that b^2 = a^2*(e^2 - 1), we can find the value of b:
b^2 = 3.4641^2 * (1.1547^2 - 1) = 4
b = 2
So the equation of the hyperbola (with vertical diretrices and center in (0,0)) is:
x^2/a^2 - y^2/b^2 = 1
x^2/12 - y^2/4 = 1
