Final answer:
The equation of the ellipse with the given conditions is (x²/192) + (y²/48) = 1, where the lengths of the major and minor axes are derived based on the focus, and the fact that the major axis is twice as long as the minor axis.
Step-by-step explanation:
To find the equation of an ellipse with center at the origin, symmetric with respect to the x- and y-axes, and given a focus at (6,0), we start by noting that the focus lies on the major axis. Since the ellipse is symmetric and centered at the origin, the other focus would be at (-6,0). The distance between the foci is 2c, so in this case, c=6.
The major axis is twice as long as the minor axis. If 2a is the length of the major axis and 2b is the length of the minor axis, then we have the relationship a=2b or b=a/2. The semi-major axis is a and the semi-minor axis is b.
The relationship between a, b, and c for any ellipse is c² = a² - b². We can substitute b = a/2 into this equation to get c² = a² - (a/2)² = a² - a²/4 = (3/4)a². Since we know that c=6, we can solve for a: 6² = (3/4)a², which gives us a² = 144 * 4/3 = 192. Therefore, a = √192. Since b = a/2, b = √192/2 = √48.
The standard form of the equation for an ellipse centered at the origin is (x²/a²) + (y²/b²) = 1. Substituting the values for a² and b², we get the equation of the ellipse: (x²/192) + (y²/48) = 1.