The equation of an ellipse is given below. (x^(2))/(15)+((y-5)^(2))/(20)=1 What are the foci of this ellipse? Choose 1 answer: (A) (0,10) and (0,0) (B) (0,5+sqrt5) and (0,5-sqrt5) (C) (5,5) and (-5,5) (D) (0+sqrt5,5) and (0-sqrt5,5)

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The equation of an ellipse is given below.  (x^(2))/(15)+((y-5)^(2))/(20)=1 What are the foci of this ellipse? Choose 1 answer: (A)  (0,10) and  (0,0) (B)  (0,5+sqrt5) and  (0,5-sqrt5) (C)  (5,5) and  (-5,5) (D)  (0+sqrt5,5) and  (0-sqrt5,5)

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Identify values of a^2 and b^2: Identify the values of a^2 and b^2 from the given standard form of the ellipse equation. 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 semi-major axis, and b is the semi-minor axis. From the given equation (x^(2))/(15)+((y-5)^(2))/(20)=1, we can see that a^2 = 20 and b^2 = 15.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 located along the y-axis at a distance of c from the center, where c is the distance from the center to a focus.Calculate value of c: Calculate the value of c using the relationship c^2 = a^2 - b^2. c^2 = 20 - 15 c^2 = 5 c = sqrt(5)Identify center of the ellipse: Identify the center of the ellipse. The center of the ellipse is at (h,k), which can be determined from the given equation. Since there is no (x-h)^2 term, h=0. The (y-k)^2 term is (y-5)^2, so k=5. Therefore, the center of the ellipse is at (0,5).Determine coordinates of the foci: Determine the coordinates of the foci. The foci are located at a distance of c from the center along the major axis. Since the major axis is vertical, the foci will have the same x-coordinate as the center, which is 0, and y-coordinates will be k ± c. So the coordinates of the foci are (0, 5 + sqrt(5)) and (0, 5 - sqrt(5)).

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