The equation of an ellipse is given below. ((x-1)^(2))/(45)+(y^(2))/(54)=1 What are the foci of this ellipse? Choose 1 answer: (A) (1,3) and (1,-3) (B) (-1,3) and (-1,-3) (C) (4,0) and (-2,0) (D) (3,-1) and (-3,-1)

Identify standard form: Identify the standard form of the ellipse equation. The given equation is already in the standard form of an ellipse, 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, and b is the semi-minor axis.Determine values of a^2, b^2, and center: Determine the values of a^2, b^2, and the center (h, k). From the given equation ((x-1)^2)/45 + (y^2)/54 = 1, we can see that a^2 = 45, b^2 = 54, and the center (h, k) = (1, 0).Identify major axis: Identify which axis is the major axis. Since b^2 > a^2, the major axis is along the y-axis.Calculate distance c: Calculate the distance c from the center to the foci. The distance c is found using the equation c^2 = b^2 - a^2. Let's calculate it. c^2 = 54 - 45 c^2 = 9 c = sqrt(9) c = 3Determine coordinates of foci: Determine the coordinates of the foci. Since the major axis is along the y-axis, the foci will be at (h, k ± c). Substituting the values we have: Foci = (1, 0 ± 3) This gives us the two foci at (1, 3) and (1, -3).

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The equation of an ellipse is given below.

((x-1)^(2))/(45)+(y^(2))/(54)=1
What are the foci of this ellipse?
Choose 1 answer:
(A)
(1,3) and
(1,-3)
(B)
(-1,3) and
(-1,-3)
(C)
(4,0) and
(-2,0)
(D)
(3,-1) and
(-3,-1)

The equation of an ellipse is given below. $\newline$ $\frac{(x-1)^{2}}{45}+\frac{y^{2}}{54}=1$ $\newline$ What are the foci of this ellipse? $\newline$ Choose $1$ answer: $\newline$ (A) $(1,3)$ and $(1,-3)$ $\newline$ (B) $(-1,3)$ and $(-1,-3)$ $\newline$ (C) $(4,0)$ and $(-2,0)$ $\newline$ (D) $(3,-1)$ and $(-3,-1)$

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Algebra 2

Find properties of ellipses from equations in general form

Full solution

Q. The equation of an ellipse is given below.

\newline

\frac{(x-1)^{2}}{45}+\frac{y^{2}}{54}=1

\newline

What are the foci of this ellipse?

\newline

Choose

1

answer:

\newline

(A)

(1,3)

and

(1,-3)

\newline

(B)

(-1,3)

and

(-1,-3)

\newline

(C)

(4,0)

and

(-2,0)

\newline

(D)

(3,-1)

and

(-3,-1)

Identify standard form: Identify the standard form of the ellipse equation. $\newline$ The given equation is already in the standard form of an ellipse, 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, and $b$ is the semi-minor axis.
Determine values of $a^2$ , $b^2$ , and center: Determine the values of $a^2$ , $b^2$ , and the center $(h, k)$ . From the given equation $\frac{(x-1)^2}{45} + \frac{y^2}{54} = 1$ , we can see that $a^2 = 45$ , $b^2 = 54$ , and the center $(h, k) = (1, 0)$ .
Identify major axis: Identify which axis is the major axis. Since b^2 > a^2, the major axis is along the y-axis.
Calculate distance $c$ : Calculate the distance $c$ from the center to the foci. $\newline$ The distance $c$ is found using the equation $c^2 = b^2 - a^2$ . Let's calculate it. $\newline$ $c^2 = 54 - 45$ $\newline$ $c^2 = 9$ $\newline$ $c = \sqrt{9}$ $\newline$ $c = 3$
Determine coordinates of foci: Determine the coordinates of the foci. $\newline$ Since the major axis is along the y-axis, the foci will be at $(h, k \pm c)$ . Substituting the values we have: $\newline$ Foci = $(1, 0 \pm 3)$ $\newline$ This gives us the two foci at $(1, 3)$ and $(1, -3)$ .