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Write an equation for an ellipse centered at the origin, which has foci at
(0,+-2) and vertices at
(0,+-sqrt15)

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm 2)$ and vertices at $(0, \pm \sqrt{15})$ .

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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

(0, \pm 2)

and vertices at

(0, \pm \sqrt{15})

.

Ellipse Orientation: We have: $\newline$ Foci: $(0, \pm 2)$ $\newline$ Vertices: $(0, \pm \sqrt{15})$ $\newline$ Choose the orientation of the ellipse. $\newline$ Since the foci and vertices are on the y-axis, the orientation is vertical.
Center and Vertex: We have: $\newline$ Center $(h, k): (0, 0)$ $\newline$ Vertices: $(0, \pm\sqrt{15})$ $\newline$ Identify the value of $a$ (the distance from the center to a vertex). $\newline$ $a = \sqrt{15}$
Foci and Center: We have: $\newline$ Center $h, k$ : $0, 0$ $\newline$ Foci: $0, \pm 2$ $\newline$ Identify the value of $c$ (the distance from the center to a focus). $\newline$ $c = 2$
Calculating $b^2$ : We know: $\newline$ $a^2 = b^2 + c^2$ (relationship between $a$ , $b$ , and $c$ in an ellipse with vertical orientation) $\newline$ We have $a = \sqrt{15}$ and $c = 2$ . $\newline$ Calculate the value of $b^2$ . $\newline$ $b^2 = a^2 - c^2$ $\newline$ $b^2 = (\sqrt{15})^2 - 2^2$ $\newline$ $a^2 = b^2 + c^2$ $0$ $\newline$ $a^2 = b^2 + c^2$ $1$
Standard Form of Ellipse: We know: $\newline$ $(h, k) = (0, 0)$ $\newline$ $a = \sqrt{15}$ $\newline$ $b^2 = 11$ $\newline$ What would be the standard form of the ellipse? $\newline$ The standard form of an ellipse with vertical orientation is: $\newline$ $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ $\newline$ Substitute the values of h, k, a, and b into the equation. $\newline$ $\frac{(x - 0)^2}{11} + \frac{(y - 0)^2}{(\sqrt{15})^2} = 1$ $\newline$ $\frac{x^2}{11} + \frac{y^2}{15} = 1$

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