The given focal points and the lengths of the minor and major axis of 16 and 20 gives the equation of the ellipse as the option;
[tex] B. \: \frac{(x + 3)^{2}}{ 100} + \frac{(y - 2)^{2}}{ 64} = 1 [/tex]
The equation of an ellipse is presented as follows;
[tex] \mathbf{ \frac{(x - h)^{2}}{ {a}^{2} } + \frac{(y - k)^{2}}{ {b}^{2} } } = 1 [/tex]
Where;
(x, y) = Coordinates of the center of the ellipse
a = Semi major axis
b = Semi minor axis
Length of minor axis = 16 units
Length of major axis = 20 units
By observation of the coordinates of the focal point, we have;
y-value of the center of the ellipse, k = 2
x-value of the center, h = (-9 + (3 - (-9))/2 = -3
The equation of the ellipse is therefore;
[tex] \frac{(x - ( - 3))^{2}}{ {10}^{2} } + \frac{(y - 2)^{2}}{ {8}^{2} } = 1 [/tex]
[tex] \frac{(x + 3)^{2}}{ 100} + \frac{(y - 2)^{2}}{ 64} = 1 [/tex]
The equation that represents the ellipse is the option;
[tex] B. \: \frac{(x + 3)^{2}}{ 100} + \frac{(y - 2)^{2}}{ 64} = 1 [/tex]
Learn more about the parts and equation of an ellipse here:
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