The equation of an ellipse is given below. ((x-4)^(2))/(9)+((y+6)^(2))/(4)=1 What are the foci of this ellipse? Choose 1 answer: (A) (4+sqrt5,-6) and (4-sqrt5,-6) (B) (13,-6) and (-5,-6) (c) (4,3) and (4,-15) (D) (4,-6+sqrt5) and (4,-6-sqrt5)

Given equation of the ellipse: Given equation of the ellipse: ((x-4)^(2))/(9)+((y+6)^(2))/(4)=1. Identify the center (h, k), the lengths of the semi-major axis (a), and the semi-minor axis (b). The center is at (h, k) = (4, -6). The length of the semi-major axis is the square root of the larger denominator, so a = √9 = 3. The length of the semi-minor axis is the square root of the smaller denominator, so b = √4 = 2.Identify the center, lengths of the semi-major axis, and semi-minor axis: Calculate the distance c from the center to the foci using the formula c = √(a^2 - b^2). Here, a = 3 and b = 2. c = √(3^2 - 2^2) = √(9 - 4) = √5.Calculate the distance c from the center to the foci: Determine the coordinates of the foci. Since the larger denominator is under the (x-4)^2 term, the foci are horizontal from the center. The foci are at (h ± c, k) = (4 ± √5, -6).Determine the coordinates of the foci: Write the final coordinates of the foci. The foci are at (4 + √5, -6) and (4 - √5, -6).

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The equation of an ellipse is given below.

((x-4)^(2))/(9)+((y+6)^(2))/(4)=1
What are the foci of this ellipse?
Choose 1 answer:
(A)
(4+sqrt5,-6) and
(4-sqrt5,-6)
(B)
(13,-6) and
(-5,-6)
(c)
(4,3) and
(4,-15)
(D)
(4,-6+sqrt5) and
(4,-6-sqrt5)

The equation of an ellipse is given below. $\newline$ $\frac{(x-4)^{2}}{9}+\frac{(y+6)^{2}}{4}=1$ $\newline$ What are the foci of this ellipse? $\newline$ Choose $1$ answer: $\newline$ (A) $(4+\sqrt{5},-6)$ and $(4-\sqrt{5},-6)$ $\newline$ (B) $(13,-6)$ and $(-5,-6)$ $\newline$ (C) $(4,3)$ and $(4,-15)$ $\newline$ (D) $(4,-6+\sqrt{5})$ and $(4,-6-\sqrt{5})$

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Math Problems

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-4)^{2}}{9}+\frac{(y+6)^{2}}{4}=1

\newline

What are the foci of this ellipse?

\newline

Choose

1

answer:

\newline

(A)

(4+\sqrt{5},-6)

and

(4-\sqrt{5},-6)

\newline

(B)

(13,-6)

and

(-5,-6)

\newline

(C)

(4,3)

and

(4,-15)

\newline

(D)

(4,-6+\sqrt{5})

and

(4,-6-\sqrt{5})

Given equation of the ellipse: Given equation of the ellipse: $\frac{(x-4)^{2}}{9}+\frac{(y+6)^{2}}{4}=1$ . Identify the center $(h, k)$ , the lengths of the semi-major axis $(a)$ , and the semi-minor axis $(b)$ . The center is at $(h, k) = (4, -6)$ . The length of the semi-major axis is the square root of the larger denominator, so $a = \sqrt{9} = 3$ . The length of the semi-minor axis is the square root of the smaller denominator, so $b = \sqrt{4} = 2$ .
Identify the center, lengths of the semi-major axis, and semi-minor axis: Calculate the distance $c$ from the center to the foci using the formula $c = \sqrt{a^2 - b^2}$ . Here, $a = 3$ and $b = 2$ . $c = \sqrt{3^2 - 2^2} = \sqrt{9 - 4} = \sqrt{5}$ .
Calculate the distance $c$ from the center to the foci: Determine the coordinates of the foci. $\newline$ Since the larger denominator is under the $(x-4)^2$ term, the foci are horizontal from the center. $\newline$ The foci are at $(h \pm c, k) = (4 \pm \sqrt{5}, -6)$ .
Determine the coordinates of the foci: Write the final coordinates of the foci. $\newline$ The foci are at $(4 + \sqrt{5}, -6)$ and $(4 - \sqrt{5}, -6)$ .