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Write the equation in standard form for the ellipse with vertices $(-7,0)$ and $(7,0)$ , and co-vertices $(0,4)$ and $(0,-4)$ .

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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

(-7,0)

and

(7,0)

, and co-vertices

(0,4)

and

(0,-4)

.

Identify Major Axis: Vertices are $(-7, 0)$ and $(7, 0)$ , so the major axis is horizontal and the center is at the origin $(0, 0)$ .
Calculate Major Axis Length: Distance between vertices is $14$ (from $-7$ to $7$ ), so the length of the major axis is $14$ . Half of this is $a$ , so $a = \frac{14}{2} = 7$ .
Identify Minor Axis: Co-vertices are $(0, 4)$ and $(0, -4)$ , so the minor axis is vertical.
Calculate Minor Axis Length: Distance between co-vertices is $8$ (from $4$ to $-4$ ), so the length of the minor axis is $8$ . Half of this is $b$ , so $b = \frac{8}{2} = 4$ .
Standard Form of Ellipse Equation: The standard form of the equation of an ellipse with a horizontal major axis is $x-h)^2/a^2 + (y-k)^2/b^2 = 1\, where \$h, k$ is the center.
Plug in Values: Plug in the values: $h = 0$ , $k = 0$ , $a = 7$ , and $b = 4$ into the equation.
Simplify Equation: The equation becomes $(x-0)^2/7^2 + (y-0)^2/4^2 = 1$ .
Simplify Equation: The equation becomes $(x-0)^2/7^2 + (y-0)^2/4^2 = 1$ . Simplify the equation to get $x^2/49 + y^2/16 = 1$ .

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