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Write an equation for an ellipse centered at the origin, which has foci at
(0,+-6) and vertices at
(0,+-sqrt37).

Write an equation for an ellipse centered at the origin, which has foci at $(0, \pm 6)$ and vertices at $(0, \pm \sqrt{37})$ .

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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 6)

and vertices at

(0, \pm \sqrt{37})

.

Given Foci and Vertices: We are given the foci at $(0, \pm 6)$ and vertices at $(0, \pm\sqrt{37})$ . Since the foci and vertices are on the y-axis, this indicates a vertical orientation for the ellipse.
Calculate $c$ : 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 value of $c$ using the coordinates of the foci. $\newline$ $c = 6$
Calculate $a$ : We can find the value of $a$ using the coordinates of the vertices. $a = \sqrt{37}$
Standard Form Equation: For an ellipse centered at the origin with a vertical orientation, the standard form of the equation is: $\newline$ $(\frac{x^2}{b^2}) + (\frac{y^2}{a^2}) = 1$ $\newline$ We have the value of $a$ , but we need to find the value of $b$ .
Relationship between $a$ , $b$ , and $c$ : The relationship between $a$ , $b$ , and $c$ for an ellipse is given by the equation: $\newline$ $c^2 = a^2 - b^2$ $\newline$ We can use this to solve for $b^2$ . $\newline$ $b^2 = a^2 - c^2$
Find $b^2$ : Substitute the known values of $a$ and $c$ into the equation to find $b^2$ .
$b^2 = (\sqrt{37})^2 - 6^2$
$b^2 = 37 - 36$
$b^2 = 1$
Write Standard Form Equation: Now that we have the values of $a$ and $b$ , we can write the standard form equation of the ellipse. $\newline$ The equation is: $\newline$ $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ $\newline$ Substitute $b^2 = 1$ and $a^2 = 37$ into the equation: $\newline$ $\frac{x^2}{1} + \frac{y^2}{37} = 1$
Simplify Equation: Simplify the equation to get the final standard form: $x^2 + \frac{y^2}{37} = 1$

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