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The equation of an ellipse is given below.

(x^(2))/(17)+(y^(2))/(35)=1
What are the foci of this ellipse?
Choose 1 answer:
(A)
(sqrt18,0) and
(-sqrt18,0)
(B)
(sqrt35,0) and
(-sqrt35,0)
(c)
(0,sqrt18) and
(0,-sqrt18)
(D)
(0,sqrt35) and
(0,-sqrt35)

The equation of an ellipse is given below. $\newline$ $\frac{x^{2}}{17}+\frac{y^{2}}{35}=1$ $\newline$ What are the foci of this ellipse? $\newline$ Choose $1$ answer: $\newline$ (A) $(\sqrt{18}, 0)$ and $(-\sqrt{18}, 0)$ $\newline$ (B) $(\sqrt{35}, 0)$ and $(-\sqrt{35}, 0)$ $\newline$ (C) $(0, \sqrt{18})$ and $(0,-\sqrt{18})$ $\newline$ (D) $(0, \sqrt{35})$ and $(0,-\sqrt{35})$

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Find properties of ellipses from equations in general form

Full solution

Q. The equation of an ellipse is given below.

\newline

\frac{x^{2}}{17}+\frac{y^{2}}{35}=1

\newline

What are the foci of this ellipse?

\newline

Choose

1

answer:

\newline

(A)

(\sqrt{18}, 0)

and

(-\sqrt{18}, 0)

\newline

(B)

(\sqrt{35}, 0)

and

(-\sqrt{35}, 0)

\newline

(C)

(0, \sqrt{18})

and

(0,-\sqrt{18})

\newline

(D)

(0, \sqrt{35})

and

(0,-\sqrt{35})

Identify lengths of axes: Identify the lengths of the semi-major and semi-minor axes. $\newline$ The standard form of an ellipse is $(\frac{x^2}{a^2}) + (\frac{y^2}{b^2}) = 1$ , where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis. In the given equation, $a^2 = 35$ and $b^2 = 17$ .
Determine major axis: Determine which axis is the major axis. $\newline$ Since $a^2 = 35$ and $b^2 = 17$ , and $a^2 > b^2$ , the major axis is along the y-axis and the minor axis is along the x-axis.
Calculate distance to foci: Calculate the distance $c$ from the center to the foci. $\newline$ The distance $c$ is found using the equation $c^2 = a^2 - b^2$ . Here, $a^2 = 35$ and $b^2 = 17$ , so $c^2 = 35 - 17 = 18$ .
Calculate value of c: Calculate the value of $c$ . Taking the square root of both sides of $c^2 = 18$ , we get $c = \sqrt{18}$ .
Identify coordinates of foci: Identify the coordinates of the foci. $\newline$ Since the major axis is along the y-axis, the foci are at $(0, \pm c)$ . Therefore, the coordinates of the foci are $(0, \sqrt{18})$ and $(0, -\sqrt{18})$ .

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