i need help with this math problem its for Ellipses.

Question

Center at the origin, symmetric with respect to the x– and y-axes, focus at (3,0), and major axis is twice as long as minor axis.

Answer 1

Let a be 1/2 the length of the major axis, b is half the length of the minor axis, and c is half the distance between the foci, or the distance from the center to the focus.

The foci lie on the major axis.

Thus, since the center is at (0, 0) and the focus is at (3, 0), x is the major axis, and c = 3 - 0 = 3

As the major axis is twice the minor axis, a = 2b

For an ellipse, √(b² + c²)= a, or b² + c² = a²

Substituting a = 2b and c = 3,

b² + 3² = (2b)²

b² + 9= 4b²

9 = 3b²

3 = b²

b = √3

a= 2b = 2√3

Then, the equation of the ellipse is

(x - h)²/a² + (y - k)²/b² = 1, where the center of the ellipse is (h, k) and a and b are as defined.

(h, k) = (0, 0), b² = (√3)² = 3, a² = (2√3)² = 12, x - h = x - 0 = x, and y - k = y - 0 = y

Thus, the equation of the ellipse is

x²/12+ y²/3 = 1