Final answer:
The equations for the given ellipses are x²/16 + y²/25 = 1 and x²/16 + y²/12 = 1, based on the lengths of their semimajor and semiminor axes as well as their foci distances.
Step-by-step explanation:
To find the equation of an ellipse, we need to identify its semimajor and semiminor axes based on the coordinates of its foci and vertices.
For the first ellipse: The vertices at (0, 5) and (0, –5) indicate the semimajor axis length is 5 units, since the vertices lie on the major axis and the distance from the center to a vertex is the semimajor axis length
(a). With the foci at (0, 3) and (0, –3), this reveals the focal distance (c) is 3 units. Using the formula c² = a² – b², where b is the semiminor axis length, we can solve for b². Thus, b² = 5² – 3² = 16. The equation of the ellipse becomes x²/16 + y²/25 = 1.
For the second ellipse: The vertices at (4, 0) and (–4, 0) indicate the semimajor axis length is 4 units. With the foci at (2, 0) and (–2, 0), we find the focal distance (c) is 2 units. Solving for b² using the formula again, we get b² = 4² – 2² = 12. Hence, the equation of the second ellipse is x²/16 + y²/12 = 1.