Answer:
(x -10)²/64 +(y +8)²/100 = 1
Explanation:
You want the equation of the ellipse with center (10,-8), a focus at (10, -14), and a vertex at (10, -18).
Axes
The length of the semi-major axis is the distance between the center and the give vertex: a = -8 -(-18) = 10 units.
The distance from the center to the focus is -8 -(-14) = 6.
The distance from the center to the covertex is the other leg of the right triangle with these distances as the hypotenuse and one leg.
b = √(10² -6²) = √64 = 8 . . . . units
Equation
The equation for the ellipse with semi-axes 'a' and 'b' with center (h, k) is ...
(x -h)²/b² +(y -k)²/a² = 1
(x -10)²/64 +(y +8)²/100 = 1
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Additional comment
The center, focus, and given vertex are all on the vertical line x=10, This means the major axis is in the vertical direction, and the denominator of the y-term will be the larger of the two denominators.
You will notice the center-focus-covertex triangle is a 3-4-5 right triangle with a scale factor of 2.