To write the equation of an ellipse in standard form with the center at the origin, we need to use the following equation:
x^2/a^2 + y^2/b^2 = 1
where (0, 0) is the center of the ellipse, a is the length of the semi-major axis (half the length of the longer axis), and b is the length of the semi-minor axis (half the length of the shorter axis).
From the given characteristics, we know that the vertex is (-5,0) which means that a = 5 (since the distance from the center to a vertex is the length of the semi-major axis). We also know that the co-vertex is (0,4) which means that b = 4 (since the distance from the center to a co-vertex is the length of the semi-minor axis).
Using these values, we can plug them into the standard form equation and simplify:
x^2/5^2 + y^2/4^2 = 1
x^2/25 + y^2/16 = 1
Therefore, the equation of the ellipse in standard form with the center at the origin and with the given characteristics is A. x^2/25 + y^2/16 = 1.