The equation of an ellipse is given below. ((x-5)^(2))/(3)+((y-7)^(2))/(6)=1 What are the foci of this ellipse? Choose 1 answer: (A) (5,7+sqrt3) and (5,7-sqrt3) (B) (-5,-7+sqrt3) and (-5,-7-sqrt3) (C) (-5+sqrt3,-7) and (-5-sqrt3,-7) (D) (5+sqrt3,7) and (5-sqrt3,7)

Question

The equation of an ellipse is given below.  ((x-5)^(2))/(3)+((y-7)^(2))/(6)=1 What are the foci of this ellipse? Choose 1 answer: (A)  (5,7+sqrt3) and  (5,7-sqrt3) (B)  (-5,-7+sqrt3) and  (-5,-7-sqrt3) (C)  (-5+sqrt3,-7) and  (-5-sqrt3,-7) (D)  (5+sqrt3,7) and  (5-sqrt3,7)

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Identify Center and Axes: Identify the center and lengths of the semi-major and semi-minor axes. The given equation of the ellipse is ((x-5)^(2))/(3)+((y-7)^(2))/(6)=1. The standard form of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h, k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis. For our ellipse, h=5, k=7, a^2=3, and b^2=6. Therefore, a=√3 and b=√6.Determine Major Axis: Determine which axis is the major axis. Since a^2 < b^2, the major axis is along the y-direction. This means that the foci will be found by moving up and down from the center along the y-axis.Calculate Distance to Foci: Calculate the distance from the center to the foci. The distance c from the center to each focus is given by the formula c^2 = b^2 - a^2. Plugging in the values we have c^2 = 6 - 3 = 3. Therefore, c = √3.Find Foci Coordinates: Find the coordinates of the foci. The foci are located at (h, k±c) since the major axis is vertical. Substituting the values we have, the foci are at (5, 7±√3).Choose Correct Answer: Choose the correct answer. The correct answer is (A) (5, 7+√3) and (5, 7-√3), since these are the coordinates we found for the foci.

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