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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

Full solution

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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