Final answer:
The equation of an ellipse with center (0, 0) and a major axis two times longer than the minor axis is x^2/4 + y^2 = 1.
Step-by-step explanation:
The equation of an ellipse with center (0, 0) can be represented as x^2/a^2 + y^2/b^2 = 1, where a is the length of the semi-major axis and b is the length of the semi-minor axis. Given that the length of the major axis is two times that of the minor axis, we can write a = 2b.
(a) x^2 + 4y^2 = 1:
The given equation is not in the standard form of an ellipse. The coefficients of x^2 and y^2 should be the squares of a and b respectively. Therefore, the equation does not represent an ellipse.
(b) x^2 + y^2 = 1:
The given equation is in the standard form of an ellipse. Comparing it with the standard equation, we have a^2 = 1 and b^2 = 1. Since a = 2b, we can solve for a and b as a = 2 and b = 1. Therefore, the correct equation of the ellipse is x^2/4 + y^2 = 1.
(c) x^2 + 4y^2 = 1:
Similar to case (a), the given equation is not in the standard form of an ellipse. Therefore, it does not represent an ellipse.
(d) 4x^2 + y^2 = 1:
The given equation is in the standard form of an ellipse. Comparing it with the standard equation, we have a^2 = 1/4 and b^2 = 1. Since a = 2b, we can solve for a and b as a = 1/2 and b = 1. Therefore, the correct equation of the ellipse is 4x^2 + y^2 = 1.