The equation of an ellipse is given below. ((x+7)^(2))/(8)+((y-10)^(2))/(13)=1 What are the foci of this ellipse? Choose 1 answer: (A) (-7+sqrt5,10) and (-7-sqrt5,10) (B) (7,-10+sqrt105) and (7,-10-sqrt105) (C) (7+sqrt105,-10) and (7-sqrt105,-10) (D) (-7,10+sqrt5) and (-7,10-sqrt5)

Identify Standard Form: Identify the standard form of the ellipse equation. The standard form of an ellipse equation 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. If a^2 > b^2, the ellipse is horizontal, and if b^2 > a^2, the ellipse is vertical.Compare Given Equation: Compare the given equation to the standard form to find h, k, a^2, and b^2. Given equation: ((x+7)^2)/8 + ((y-10)^2)/13 = 1 Standard form: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 Comparing the two, we find: h = -7, k = 10, a^2 = 13, b^2 = 8Determine Major Axis: Determine which axis is the major axis. Since a^2 > b^2, the major axis is along the y-axis, and the minor axis is along the x-axis.Calculate Distance to Foci: Calculate the distance from the center to the foci along the major axis. The distance c from the center to each focus is given by c = sqrt(a^2 - b^2). Here, a^2 = 13 and b^2 = 8, so c = sqrt(13 - 8) = sqrt(5).Find Foci Coordinates: Find the coordinates of the foci. Since the major axis is vertical, the foci are located at (h, k ± c). Using h = -7, k = 10, and c = sqrt(5), we get the foci at: (-7, 10 + sqrt(5)) and (-7, 10 - sqrt(5)).

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The equation of an ellipse is given below.

((x+7)^(2))/(8)+((y-10)^(2))/(13)=1
What are the foci of this ellipse?
Choose 1 answer:
(A)
(-7+sqrt5,10) and
(-7-sqrt5,10)
(B)
(7,-10+sqrt105) and
(7,-10-sqrt105)
(C)
(7+sqrt105,-10) and
(7-sqrt105,-10)
(D)
(-7,10+sqrt5) and
(-7,10-sqrt5)

The equation of an ellipse is given below. $\newline$ $\frac{(x+7)^{2}}{8}+\frac{(y-10)^{2}}{13}=1$ $\newline$ What are the foci of this ellipse? $\newline$ Choose $1$ answer: $\newline$ (A) $(-7+\sqrt{5}, 10)$ and $(-7-\sqrt{5}, 10)$ $\newline$ (B) $(7,-10+\sqrt{105})$ and $(7,-10-\sqrt{105})$ $\newline$ (C) $(7+\sqrt{105},-10)$ and $(7-\sqrt{105},-10)$ $\newline$ (D) $(-7,10+\sqrt{5})$ and $(-7,10-\sqrt{5})$

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Algebra 1

Compare linear, exponential, and quadratic growth

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Q. The equation of an ellipse is given below.

\newline

\frac{(x+7)^{2}}{8}+\frac{(y-10)^{2}}{13}=1

\newline

What are the foci of this ellipse?

\newline

Choose

1

answer:

\newline

(A)

(-7+\sqrt{5}, 10)

and

(-7-\sqrt{5}, 10)

\newline

(B)

(7,-10+\sqrt{105})

and

(7,-10-\sqrt{105})

\newline

(C)

(7+\sqrt{105},-10)

and

(7-\sqrt{105},-10)

\newline

(D)

(-7,10+\sqrt{5})

and

(-7,10-\sqrt{5})

Identify Standard Form: Identify the standard form of the ellipse equation. $\newline$ The standard form of an ellipse equation 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. If a^2 > b^2, the ellipse is horizontal, and if b^2 > a^2, the ellipse is vertical.
Compare Given Equation: Compare the given equation to the standard form to find $h$ , $k$ , $a^2$ , and $b^2$ .
Given equation: $\frac{(x+7)^2}{8} + \frac{(y-10)^2}{13} = 1$
Standard form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
Comparing the two, we find:
$h = -7$ , $k = 10$ , $a^2 = 13$ , $b^2 = 8$
Determine Major Axis: Determine which axis is the major axis. $\newline$ Since a^2 > b^2, the major axis is along the y-axis, and the minor axis is along the x-axis.
Calculate Distance to Foci: Calculate the distance from the center to the foci along the major axis. $\newline$ The distance $c$ from the center to each focus is given by $c = \sqrt{a^2 - b^2}$ . $\newline$ Here, $a^2 = 13$ and $b^2 = 8$ , so $c = \sqrt{13 - 8} = \sqrt{5}$ .
Find Foci Coordinates: Find the coordinates of the foci. $\newline$ Since the major axis is vertical, the foci are located at $(h, k \pm c)$ . $\newline$ Using $h = -7$ , $k = 10$ , and $c = \sqrt{5}$ , we get the foci at: $\newline$ $(-7, 10 + \sqrt{5})$ and $(-7, 10 - \sqrt{5})$ .