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Write an equation for an ellipse centered at the origin, which has foci at
(0,+-sqrt63) and vertices at
(0,+-sqrt91).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm \sqrt{63})$ and vertices at $(0, \pm \sqrt{91})$ .

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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{63})

and vertices at

(0, \pm \sqrt{91})

.

Given Points and Orientation: We are given the foci at $(0, \pm\sqrt{63})$ and vertices at $(0, \pm\sqrt{91})$ . Since the foci and vertices are on the y-axis, this indicates a vertical orientation for the ellipse.
Calculate $c$ and $a$ : The distance from the center to a focus is denoted by $c$ , and the distance from the center to a vertex is denoted by $a$ . We can find the values of $c$ and $a$ using the coordinates of the foci and vertices, respectively. $\newline$ $c = \sqrt{63}$ $\newline$ $a = \sqrt{91}$
Calculate $b$ : We can find the value of $b$ , which represents the distance from the center to the co-vertices, by using the relationship $a^2 = b^2 + c^2$ for ellipses with vertical major axes. $\newline$ $b^2 = a^2 - c^2$ $\newline$ $b^2 = (\sqrt{91})^2 - (\sqrt{63})^2$ $\newline$ $b^2 = 91 - 63$ $\newline$ $b^2 = 28$ $\newline$ $b = \sqrt{28}$
Standard Form of Ellipse: Now that we have the values for $a$ and $b$ , we can write the equation of the ellipse in standard form. For an ellipse centered at the origin $(h, k)$ with a vertical major axis, the standard form is: $\newline$ $(x - h)^2/b^2 + (y - k)^2/a^2 = 1$ $\newline$ Since the center is at the origin $(0, 0)$ , $h = 0$ and $k = 0$ , and the equation simplifies to: $\newline$ $x^2/b^2 + y^2/a^2 = 1$
Substitute Values: Substitute the values of $a$ and $b$ into the equation: $\newline$ $\frac{x^2}{\sqrt{28}^2} + \frac{y^2}{\sqrt{91}^2} = 1$ $\newline$ $\frac{x^2}{28} + \frac{y^2}{91} = 1$

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