The equation of an ellipse is given below. $\newline$ $\frac{(x-5)^{2}}{3}+\frac{(y-7)^{2}}{6}=1$ $\newline$ What are the foci of this ellipse? $\newline$ Choose $1$ answer: $\newline$ (A) $(5,7+\sqrt{3})$ and $(5,7-\sqrt{3})$ $\newline$ (B) $(-5,-7+\sqrt{3})$ and $(-5,-7-\sqrt{3})$ $\newline$ (C) $(-5+\sqrt{3},-7)$ and $(-5-\sqrt{3},-7)$ $\newline$ (D) $(5+\sqrt{3}, 7)$ and $(5-\sqrt{3}, 7)$

Full solution

Q. The equation of an ellipse is given below.

\newline

\frac{(x-5)^{2}}{3}+\frac{(y-7)^{2}}{6}=1

\newline

What are the foci of this ellipse?

\newline

Choose

1

answer:

\newline

(A)

(5,7+\sqrt{3})

and

(5,7-\sqrt{3})

\newline

(B)

(-5,-7+\sqrt{3})

and

(-5,-7-\sqrt{3})

\newline

(C)

(-5+\sqrt{3},-7)

and

(-5-\sqrt{3},-7)

\newline

(D)

(5+\sqrt{3}, 7)

and

(5-\sqrt{3}, 7)

Identify Center and Axes: Identify the center and lengths of the semi-major and semi-minor axes. $\newline$ The given equation of the ellipse is $\frac{(x-5)^{2}}{3}+\frac{(y-7)^{2}}{6}=1$ . The standard form of an ellipse is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h, k)$ is the center of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis. For our ellipse, $h=5$ , $k=7$ , $a^2=3$ , and $b^2=6$ . Therefore, $a=\sqrt{3}$ and $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ $0$ .
Determine Major Axis: Determine which axis is the major axis. $\newline$ Since $a^2 < b^2$ , the major axis is along the y-direction. This means that the foci will be found by moving up and down from the center along the y-axis.
Calculate Distance to Foci: Calculate the distance from the center to the foci. The distance $c$ from the center to each focus is given by the formula $c^2 = b^2 - a^2$ . Plugging in the values we have $c^2 = 6 - 3 = 3$ . Therefore, $c = \sqrt{3}$ .
Find Foci Coordinates: Find the coordinates of the foci. $\newline$ The foci are located at $(h, k\pm c)$ since the major axis is vertical. Substituting the values we have, the foci are at $(5, 7\pm\sqrt{3})$ .
Choose Correct Answer: Choose the correct answer. $\newline$ The correct answer is (A) $(5, 7+\sqrt{3})$ and $(5, 7-\sqrt{3})$ , since these are the coordinates we found for the foci.