Final answer:
The standard form of the ellipse with y-intercepts (0, ±5) and foci (±4,0) is (x^2/41) + (y^2/25) = 1, where the semi-minor axis b is 5 and the semi-major axis a is the square root of 41 (√41).
Step-by-step explanation:
An ellipse has two fixed points called foci, and the sum of the distances from any point on the ellipse to the foci is constant. This property helps to determine the standard form of the equation of an ellipse. Given the y-intercepts are at (0, ±5) and the foci are at (±4, 0), we can find the lengths of the semi-major axis, a, and the semi-minor axis, b, and the distance between the foci, 2c, to describe the ellipse.
The semi-minor axis b is the distance from the center to the y-intercept, so b=5. The distance between the foci is 2c; since the foci are at (±4, 0), we have c=4. The semi-major axis a can be found using the relationship a² = b² + c². Plugging in the values, a² = 5² + 4² = 25 + 16 = 41, so a = √41.
The standard form of the equation of an ellipse with a horizontal major axis is given by (x2/a2) + (y2/b2) = 1. Substituting the values for a and b, the equation becomes (x2/41) + (y2/25) = 1.