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Write the equation in standard form for the ellipse with vertices $(0,12)$ and $(0,-12)$ , and co-vertices $(-4,0)$ and $(4,0)$ .

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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

(0,12)

and

(0,-12)

, and co-vertices

(-4,0)

and

(4,0)

.

Identify Vertical Ellipse: Vertices are $(0,12)$ and $(0,-12)$ , so this is a vertical ellipse. The center is at $(0,0)$ because it's the midpoint of the vertices.
Calculate $a$ Value: The distance from the center to a vertex is the value of $a$ . So, $a = 12$ because the vertex is $12$ units away from the center on the y-axis.
Calculate $b$ Value: Co-vertices are $(-4,0)$ and $(4,0)$ , so $b$ is the distance from the center to a co-vertex. That means $b = 4$ , since the co-vertex is $4$ units away from the center on the x-axis.
Plug in Values: Now we plug in the values for ' $a$ ' and ' $b$ ' into the standard form equation of an ellipse. Since it's vertical, the ' $a$ ' value goes under the $y$ -term. The equation is $(x-h)^2/b^2 + (y-k)^2/a^2 = 1$ , where $(h,k)$ is the center.
Substitute into Equation: Substitute $h=0$ , $k=0$ , $a=12$ , and $b=4$ into the equation. We get $(x-0)^2/4^2 + (y-0)^2/12^2 = 1$ .
Simplify Equation: Simplify the equation to get $x^2/16 + y^2/144 = 1$ . This is the standard form of the ellipse.

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