Answer
Thus the equation is $\frac{(x-2)^{2}}{100} + \frac{(y+3)^{2}}{64} = 1$
Work Step by Step
General Equation of an ellipse is: $\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1$
At Center (2, -3), h = 2 and k = -3
The length from (12,-3) to (2,-3) is 10, thus a (vertice) = 10
The length from (8,-3) to (2,-3) is 6, thus c (foci) = 6
Also, the relationship of c to a and b is $c^{2} = a^{2} - b^{2}$
Thus $6^{2} = 10^{2} - b^{2}$
$36 = 100 - b^{2}$
$b^{2} = 64$
Thus the equation is $\frac{(x-2)^{2}}{100} + \frac{(y+3)^{2}}{64} = 1$
