Finding the Equation of an Ellipse

Question

How do I find the equation of an ellipse given the foci and eccentricity as in the following:

Write the equation of the ellipse with foci (3,0) and (3,4) and e = .5

Answer 1

We can calculate the distance between foci (3,0) (3,4) as 4
so we know that c=2

In the formula e=c/a we know that e is .5 or 1/2 and the c value is 2

We can now easily calculate the a value as follows

1/2 = 2/a
1*a=2*2
a=4

b=√(a²-c²) = √(4²-2²) = √(16-4) =√12 = 2√3

This is a vertical ellipse center (3,2)
equation
(x - 3) ² / 12 + (y - 2)² / 16 = 1
or
4x² + 3y² - 24x - 12y = 0

Answer 2

(y-2)^2/9-(x^2)/25=1