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Write an equation for an ellipse centered at the origin, which has foci at
(0,+-sqrt55) and vertices at
(0,+-sqrt89).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm \sqrt{55})$ and vertices at $(0, \pm \sqrt{89})$ .

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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{55})

and vertices at

(0, \pm \sqrt{89})

.

Ellipse Orientation: We have: $\newline$ Foci: $(0, \pm\sqrt{55})$ $\newline$ Vertices: $(0, \pm\sqrt{89})$ $\newline$ Since the foci and vertices are on the y-axis, the ellipse is vertical.
Finding the Value of c: Identify the value of c (the distance from the center to a focus) using the coordinates of the foci. $\newline$ $c = \sqrt{55}$
Finding the Value of $a$ : Identify the value of $a$ (the distance from the center to a vertex) using the coordinates of the vertices. $\newline$ $a = \sqrt{89}$
Finding the Value of b: Use the relationship $c^2 = a^2 - b^2$ to find the value of $b^2$ (the square of the distance from the center to a co-vertex). $\newline$ $c^2 = a^2 - b^2$ $\newline$ $(\sqrt{55})^2 = (\sqrt{89})^2 - b^2$ $\newline$ $55 = 89 - b^2$ $\newline$ $b^2 = 89 - 55$ $\newline$ $b^2 = 34$
Standard Form of the Equation: Now we have: $\newline$ $a = \sqrt{89}$ $\newline$ $b = \sqrt{34}$ $\newline$ $c = \sqrt{55}$ $\newline$ The standard form of the equation for a vertical ellipse centered at the origin is: $\newline$ $(x - h)^2/b^2 + (y - k)^2/a^2 = 1$ $\newline$ Since the center $(h, k)$ is at the origin $(0, 0)$ , the equation simplifies to: $\newline$ $x^2/b^2 + y^2/a^2 = 1$
Substituting Values into the Equation: Substitute the values of $a$ and $b$ into the equation. $\newline$ $\frac{x^2}{(\sqrt{34})^2} + \frac{y^2}{(\sqrt{89})^2} = 1$ $\newline$ $\frac{x^2}{34} + \frac{y^2}{89} = 1$

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