Determine the equation of the ellipse with foci left bracket, 10, comma, minus, 6, right bracket(10,−6) and left bracket, minus, 2, comma, minus, 6, right bracket(−2,−6), and vertices left bracket, 14, comma, minus, 6, right bracket(14,−6) and left bracket, minus, 6, comma, minus, 6, right bracket(−6,−6).

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 give the same result: (10+(-2))/2 = 4 and (-6+(-6))/2 = -6. So the center is (4,-6).

2. Find the length of the semi-major axis, a. This is half the distance between the vertices. The distance between the vertices (14,-6) and (-6,-6) is 14 - (-6) = 20, so a = 20/2 = 10.

3. Find the length of the semi-minor axis, b. We use the relationship c² = a² - b², where c is the distance from the center to a focus. The distance from the center (4,-6) to a focus (10,-6) is 10 - 4 = 6, so c = 6. Substituting the known values, we get 6² = 10² - b², which simplifies to 36 = 100 - b². Solving for b² gives b² = 100 - 36 = 64, so b = sqrt(64) = 8.

4. Substitute h = 4, k = -6, a = 10, and b = 8 into the standard form equation to get the equation of the ellipse: (x-4)²/10² + (y+6)²/8² = 1. This simplifies to (x-4)²/100 + (y+6)²/64 = 1.