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Write the equation in standard form for the ellipse with vertices $(-3,0)$ and $(3,0)$ , and co-vertices $(0,2)$ and $(0,-2)$ .

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Write equations of ellipses in standard form using properties

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Q. Write the equation in standard form for the ellipse with vertices

(-3,0)

and

(3,0)

, and co-vertices

(0,2)

and

(0,-2)

.

Identify Major Axis: Vertices are $(-3,0)$ and $(3,0)$ , so the major axis is horizontal. The distance from the center to a vertex is the value of $a$ . Since the vertices are $3$ units away from the center, $a = 3$ .
Identify Minor Axis: Co-vertices are $(0,2)$ and $(0,-2)$ , so the minor axis is vertical. The distance from the center to a co-vertex is the value of $b$ . Since the co-vertices are $2$ units away from the center, $b = 2$ .
Find Center: The center $(h,k)$ is at the midpoint of the vertices, which is $(0,0)$ because the vertices are equidistant from the origin along the $x$ -axis.
Standard Form Equation: The standard form of the equation for an ellipse with a horizontal major axis is $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$ . Plugging in the values for $h$ , $k$ , $a$ , and $b$ , we get $(x-0)^2/3^2 + (y-0)^2/2^2 = 1$ .
Simplify Equation: Simplify the equation to get $x^2/9 + y^2/4 = 1$ . This is the standard form of the equation for the ellipse.

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