Determine the equation of the ellipse with foci left bracket, minus, 5, comma, 5, right bracket(−5,5) and left bracket, minus, 5, comma, minus, 19, right bracket(−5,−19), and vertices left bracket, minus, 5, comma, 6, right bracket(−5,6) and left bracket, minus, 5, comma, minus, 20, right bracket(−5,−20).

The equation of an ellipse in standard form is given by (x-h)²/a² + (y-k)²/b² = 1, where (h,k) is the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis.

1. Find the center of the ellipse. The center is the midpoint of the line segment joining the foci or the vertices. Here, both the foci and the vertices have the same x-coordinate, -5, so the x-coordinate of the center is -5. The y-coordinate of the center is the average of the y-coordinates of the foci or the vertices. So, the center of the ellipse is (-5, (5-19)/2) = (-5, -7).

2. Find the length of the major axis. The major axis is the line segment joining the vertices. Its length is the distance between the vertices, which is |6 - (-20)| = 26. Therefore, the semi-major axis a is 26/2 = 13.

3. Find the length of the minor axis. The minor axis is perpendicular to the major axis and passes through the center. Its length is the distance between the foci, which is |5 - (-19)| = 24. Therefore, the semi-minor axis b is 24/2 = 12.

4. Substitute h = -5, k = -7, a = 13, and b = 12 into the equation of the ellipse to get (x+5)²/13² + (y+7)²/12² = 1.