Final answer:
The equation of the ellipse with foci at (10,5) and (10,-11) and a major axis of length 20 is (x-10)²/52 + (y+3)²/64 =1.
Step-by-step explanation:
The distance between the foci is 16 units from 5 to-11, which is the length of the major axis. The center of the ellipse lies at the midpoint of the foci, giving the center as (10, -3).
The formula for the distance between the foci and the center in the case of an ellipse is 2c = 16, hence c = 8.
The length of the major axis is 2a = 20, therefore a = 10.
With the center at (h, k) = (10, -3), the semi-major axis a = 10 and the distance from the center to a focus c = 8, the equation for the ellipse is (x-10)²/52 + (y+3)²/64 =1.
To find the equation of the ellipse, start by identifying key information: the foci coordinates, length of the major axis, and the relationship between foci and major axis. First, calculate the distance between the foci, giving the major axis length. The center of the ellipse is at the midpoint of the foci. Then, apply the formula for the distance between the foci and the center to determine 'c'. Next, using the length of the major axis, derive the value of 'a'. Finally, substitute the values of the center, 'a', and 'c' into the standard equation of an ellipse to obtain the required equation: (x-10)²/52 + (y+3)²/64 =1.