Final answer:
The correct equation for the ellipse with the given vertices at (0, 5) and (0, -5) and foci at (0, 3) and (0, -3) is x²/16 + y²/25 = 1, which corresponds to option B.
Step-by-step explanation:
To determine the equation of the ellipse given the vertices at (0, 5) and (0, -5), and the foci at (0, 3) and (0, -3), we need to understand a few properties of ellipses. Firstly, an ellipse is a closed curve wherein the sum of the distances from any point on the curve to the two foci is constant. This means that the major axis runs vertically along the y-axis since the vertices are aligned in that direction.
The distance between the vertices, which is 10 in this case (from -5 to 5), represents the length of the major axis (2a). Hence, a = 5. The distance between the foci is 6 (from -3 to 3), which gives us the value for 2c, so c = 3. Using the relationship c² = a² - b², we can find b² since we already have c (3) and a (5).
Substituting the values into the equation we get:
9 = 25 - b²
b² = 25 - 9
b² = 16
Since the major axis is vertical, the equation of the ellipse is of the form x²/b² + y²/a² = 1. Substituting a² and b² we finally get x²/16 + y²/25 = 1, which corresponds to option B.
This is the correct equation for the ellipse with the given vertices and foci.