The equation of an ellipse is given below. ((x+15)^(2))/(676)+((y-4)^(2))/(100)=1 What are the foci of this ellipse? Choose 1 answer: A) (-15+sqrt24,4) and (-15-sqrt24,4) (B) (-15,4+sqrt24) and (-15,4-sqrt24) (c) (-39,4) and (9,4) (D) (-15,28) and (-15,-20)

Question

The equation of an ellipse is given below.  ((x+15)^(2))/(676)+((y-4)^(2))/(100)=1 What are the foci of this ellipse? Choose 1 answer: A)  (-15+sqrt24,4) and  (-15-sqrt24,4) (B)  (-15,4+sqrt24) and  (-15,4-sqrt24) (c)  (-39,4) and  (9,4) (D)  (-15,28) and  (-15,-20)

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Identify Ellipse Equation Form: Identify the standard form of the ellipse equation. The given equation is in the standard form of an ellipse: ((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.Determine Ellipse Center: Determine the center (h, k) of the ellipse. From the given equation, we can see that h = -15 and k = 4. So the center of the ellipse is (-15, 4).Identify a^2 and b^2: Identify a^2 and b^2. In the given equation, a^2 = 676 and b^2 = 100. Therefore, a = sqrt(676) = 26 and b = sqrt(100) = 10.Determine Ellipse Orientation: Determine the orientation of the ellipse. Since a^2 > b^2, the ellipse is oriented along the x-axis, which means the foci will be to the left and right of the center along the x-axis.Calculate Distance to Foci: Calculate the distance c from the center to each focus. The distance c is found using the formula c = sqrt(a^2 - b^2). Plugging in the values, we get c = sqrt(676 - 100) = sqrt(576) = 24.Find Foci Coordinates: Find the coordinates of the foci. The foci are located at (h ± c, k). Substituting the values, we get the foci at (-15 ± 24, 4). This gives us two points: (-15 + 24, 4) and (-15 - 24, 4), which simplify to (9, 4) and (-39, 4).

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