Hello,

the center of the ellipse is the midpoint between the foci: C(1, 1)

I assume that the axes of the ellipse are parallel to the coordinate axes, otherwise your problem can't be done:

Let a be the major axis and let be b the minor axis.

You know that the distance between center and the focus on the major axis is called excentricity e and that it is calculated by:

e² = a² - b² . Thus e = 4 - 1 = 3

e² + b² = a². With your problem:

9 + 64 = 73 = a². a = √(73)

Therefore the equation of your ellipse becomes:

x²/73 + y²/64 = 1

The midpoint of all vertices is the center of the ellips: C(0, 0)

The length of the axes is the distance between center and vertices. Therefore

a = 4, b = 5

The equation of the ellipse becomes:

x²/16 + y²/25 = 1

Firts: Complete the squares:

2x² + 3y² + 12x - 24y + 60 = 0

2(x² + 6x + 9 - 9) +3(y² - 8y + 16 - 16) = -60

2(x+3)² - 18 + 3(y-4)² - 48 = -60

2(x+3)² + 3(y-4)² = 6

(x+3)²/3 + (y-4)²/2 = 1

The center of the ellipse is C(-3, 4)

The major axis is a = √3; the minor axis is b = √2

e² = a² - b² ==> e² = 3 - 2 = 1

Therefore F1(3+1, 4) and F2(3-1, 4)