Write an equation for an ellipse centered at the origin, which has foci at $( \pm \sqrt{8}, 0)$ and co-vertices at $(0, \pm \sqrt{10})$ .

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Algebra 2

Write equations of ellipses in standard form using properties

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Q. Write an equation for an ellipse centered at the origin, which has foci at

( \pm \sqrt{8}, 0)

and co-vertices at

(0, \pm \sqrt{10})

Given Information: We are given the foci at $(\pm\sqrt{8}, 0)$ and co-vertices at $(0, \pm\sqrt{10})$ . The center of the ellipse is at the origin $(0, 0)$ . To write the equation of the ellipse in standard form, we need to find the values of $a$ , $b$ , and $c$ , where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $c$ is the distance from the center to a focus.
Finding the Value of $c$ : The distance from the center to a focus, $c$ , is given as $\sqrt{8}$ . Therefore, $c = \sqrt{8}$ .
Finding the Value of $b$ : The length of the semi-minor axis, $b$ , is given by the distance from the center to a co-vertex. Since the co-vertices are at $(0, \pm\sqrt{10})$ , $b = \sqrt{10}$ .
Finding the Value of a: To find the length of the semi-major axis, $a$ , we use the relationship $c^2 = a^2 - b^2$ , which comes from the definition of an ellipse. We already know $c$ and $b$ , so we can solve for $a$ .
$c^2 = a^2 - b^2$
$(\sqrt{8})^2 = a^2 - (\sqrt{10})^2$
$8 = a^2 - 10$
$a^2 = 8 + 10$
$a^2 = 18$
$c^2 = a^2 - b^2$ $0$
$c^2 = a^2 - b^2$ $1$
Writing the Equation in Standard Form: Now that we have $a$ , $b$ , and $c$ , we can write the equation of the ellipse in standard form. The standard form of an ellipse with a horizontal major axis is $(x^2/a^2) + (y^2/b^2) = 1$ .
Writing the Equation in Standard Form: Now that we have $a$ , $b$ , and $c$ , we can write the equation of the ellipse in standard form. The standard form of an ellipse with a horizontal major axis is $(x^2/a^2) + (y^2/b^2) = 1$ .Substitute the values of $a$ and $b$ into the standard form equation of the ellipse: $\newline$ $(x^2/a^2) + (y^2/b^2) = 1$ $\newline$ $(x^2/(3\sqrt{2})^2) + (y^2/(\sqrt{10})^2) = 1$ $\newline$ $(x^2/18) + (y^2/10) = 1$