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Write an equation for an ellipse centered at the origin, which has foci at
(0,+-15) and co-vertices at
(+-8,0).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm 15)$ and co-vertices at $( \pm 8,0)$ .

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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 15)

and co-vertices at

( \pm 8,0)

.

Ellipse Orientation: We have: $\newline$ Foci: $(0, \pm 15)$ $\newline$ Co-vertices: $(\pm 8, 0)$ $\newline$ Since the foci are on the $y$ -axis, this indicates a vertical orientation for the ellipse.
Identify $c$ : We have: $\newline$ Center $(h, k)$ : $(0, 0)$ $\newline$ Foci: $(0, \pm 15)$ $\newline$ Identify the value of $c$ (the distance from the center to a focus). $\newline$ $c = \sqrt{(0 - 0)^2 + (15 - 0)^2}$ $\newline$ $= \sqrt{0^2 + 15^2}$ $\newline$ $= \sqrt{225}$ $\newline$ $= 15$
Identify $b$ : We have: $\newline$ Center $(h, k)$ : $(0, 0)$ $\newline$ Co-vertices: $(\pm 8, 0)$ $\newline$ Identify the value of $b$ (the distance from the center to a co-vertex). $\newline$ $b = \sqrt{(8 - 0)^2 + (0 - 0)^2}$ $\newline$ $= \sqrt{8^2 + 0^2}$ $\newline$ $= \sqrt{64}$ $\newline$ $= 8$
Calculate $a$ : We know the relationship between $a$ , $b$ , and $c$ for an ellipse with a vertical orientation is $c^2 = a^2 - b^2$ . We have already found $c = 15$ and $b = 8$ . Now we solve for $a$ . $a^2 = c^2 + b^2 = 15^2 + 8^2 = 225 + 64 = 289$ $a = \sqrt{289} = 17$
Standard Form: We know: $\newline$ $(h, k) = (0, 0)$ $\newline$ $a = 17$ $\newline$ $b = 8$ $\newline$ The standard form of the ellipse with a 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 $b$ : $\newline$ $\frac{(x - 0)^2}{8^2} + \frac{(y - 0)^2}{17^2} = 1$ $\newline$ $\frac{x^2}{64} + \frac{y^2}{289} = 1$

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