The equation of an ellipse is given below. (x^(2))/(46)+((y+8)^(2))/(26)=1 What are the foci of this ellipse? Choose 1 answer: (A) (0+sqrt20,8) and (0-sqrt20,8) (B) (0,8+sqrt20) and (0,8-sqrt20) (C) (0,-8+sqrt20) and (0,-8-sqrt20) (D) (0+sqrt20,-8) and (0-sqrt20,-8)

Question

The equation of an ellipse is given below.  (x^(2))/(46)+((y+8)^(2))/(26)=1 What are the foci of this ellipse? Choose 1 answer: (A)  (0+sqrt20,8) and  (0-sqrt20,8) (B)  (0,8+sqrt20) and  (0,8-sqrt20) (C)  (0,-8+sqrt20) and  (0,-8-sqrt20) (D)  (0+sqrt20,-8) and  (0-sqrt20,-8)

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Given Equation of the Ellipse: The given equation of the ellipse is $(x^2)/46 + ((y+8)^2)/26 = 1$. To find the foci, we need to identify the major and minor axes and their lengths.Standard Form of an Ellipse: The standard form of an ellipse is $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$ for a horizontal ellipse, or $(x-h)^2/b^2 + (y-k)^2/a^2 = 1$ for a vertical ellipse, where (h, k) is the center of the ellipse, $a$ is the semi-major axis length, and $b$ is the semi-minor axis length. The larger denominator corresponds to the semi-major axis length squared.Center and Axis Lengths: In the given equation, the center of the ellipse is at (0, -8) because the equation can be rewritten as $(x-0)^2/46 + (y-(-8))^2/26 = 1$. The semi-major axis length squared is the larger denominator, which is 46, and the semi-minor axis length squared is the smaller denominator, which is 26.Calculate Semi-Major Axis Length: Calculate the semi-major axis length $a$ by taking the square root of 46: $a = \sqrt{46}$.Calculate Semi-Minor Axis Length: Calculate the semi-minor axis length $b$ by taking the square root of 26: $b = \sqrt{26}$.Calculate Distance to Foci: The distance from the center to the foci along the major axis is given by $c$, where $c^2 = a^2 - b^2$. Calculate $c$: $c^2 = 46 - 26$, $c^2 = 20$, $c = \sqrt{20}$.Orientation and Foci Coordinates: Since the larger denominator is under the $x^2$ term, the major axis is horizontal, and the foci will be to the left and right of the center along the x-axis. The coordinates of the foci are therefore $(h \pm c, k)$, where (h, k) is the center of the ellipse.Substitute Values for Foci Coordinates: Substitute the values of $h$, $k$, and $c$ into the foci coordinates: $h = 0$, $k = -8$, $c = \sqrt{20}$, Foci: $(0 \pm \sqrt{20}, -8)$.Foci of the Ellipse: The foci of the ellipse are $(0 + \sqrt{20}, -8)$ and $(0 - \sqrt{20}, -8)$, which corresponds to answer choice (D).

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