The equation of an ellipse is given below. ((x+9)^(2))/(100)+(y^(2))/(64)=1 What are the foci of this ellipse? Choose 1 answer: (A) (-9,-3) and (-9,3) (B) (-9,-6) and (-9,6) (C) (-15,0) and (-3,0) (D) (-12,0) and (-6,0)

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The equation of an ellipse is given below.  ((x+9)^(2))/(100)+(y^(2))/(64)=1 What are the foci of this ellipse? Choose 1 answer: (A)  (-9,-3) and  (-9,3) (B)  (-9,-6) and  (-9,6) (C)  (-15,0) and  (-3,0) (D)  (-12,0) and  (-6,0)

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Identify center and lengths: Identify the center and lengths of the major and minor axes. The given equation of the ellipse is ((x+9)^(2))/(100)+(y^(2))/(64)=1. This is in the standard form of an ellipse equation, which 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 length, and b is the semi-minor axis length. For our ellipse, h = -9, k = 0, a^2 = 100, and b^2 = 64. Therefore, a = 10 and b = 8.Determine major axis: Determine which axis is the major axis. Since a^2 > b^2, the major axis is along the x-axis, and the minor axis is along the y-axis. This means that the foci will be located along the x-axis, at a distance of c from the center, where c is found using the formula c = sqrt(a^2 - b^2).Calculate distance to foci: Calculate the distance c from the center to each focus. Using the formula c = sqrt(a^2 - b^2), we find c = sqrt(100 - 64) = sqrt(36) = 6. This means the foci are located 6 units to the left and right of the center along the x-axis.Find coordinates of foci: Find the coordinates of the foci. The center of the ellipse is at (-9, 0). The foci are located at (-9 - c, 0) and (-9 + c, 0). Substituting the value of c, we get the foci at (-9 - 6, 0) and (-9 + 6, 0), which simplifies to (-15, 0) and (-3, 0).

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