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Write an equation for an ellipse centered at the origin, which has foci at
(0,+-sqrt15) and co-vertices at
(+-sqrt21,0).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm \sqrt{15})$ and co-vertices at $( \pm \sqrt{21}, 0)$ .

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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 \sqrt{15})

and co-vertices at

( \pm \sqrt{21}, 0)

.

Given Foci and Co-vertices: We are given the foci at $(0,\pm\sqrt{15})$ and co-vertices at $(\pm\sqrt{21},0)$ . Since the foci are on the y-axis, this indicates that the ellipse is oriented vertically.
Calculating $c$ : The distance between the center and a focus is the value of $c$ in the ellipse equation. We can calculate $c$ using the coordinates of the foci. $c = \sqrt{15}$
Calculating $b$ : The distance between the center and a co-vertex is the value of $b$ in the ellipse equation. We can calculate $b$ using the coordinates of the co-vertices. $b = \sqrt{21}$
Finding a: We know that for an ellipse centered at the origin, the relationship between $a$ , $b$ , and $c$ is given by $c^2 = a^2 - b^2$ . We can use this to find the value of $a$ .
$c^2 = a^2 - b^2$
$(\sqrt{15})^2 = a^2 - (\sqrt{21})^2$
$15 = a^2 - 21$
$a^2 = 15 + 21$
$a^2 = 36$
$b$ $0$
$b$ $1$
Writing the Equation of the Ellipse: Now that we have the values of $a$ and $b$ , we can write the equation of the ellipse in standard form. Since the ellipse is vertically oriented, $a^2$ will be under the $y^2$ term and $b^2$ will be under the $x^2$ term. $\newline$ The standard form of the ellipse is: $\newline$ $(x - h)^2/b^2 + (y - k)^2/a^2 = 1$ $\newline$ Plugging in the values for $h$ , $k$ , $a$ , and $b$ , we get: $\newline$ $b$ $1$ $\newline$ $b$ $2$

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