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Write the equation in standard form for the ellipse with vertices $(0,10)$ and $(0,-10)$ , and co-vertices $(-2,0)$ and $(2,0)$ .

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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

(0,10)

and

(0,-10)

, and co-vertices

(-2,0)

and

(2,0)

.

Identify Major Axis: Vertices are $(0,10)$ and $(0,-10)$ , so the major axis is vertical. The distance from the center to a vertex is the value of ' $a$ '. Calculate ' $a$ ' using the distance formula or by observation since the center is at the origin $(0,0)$ . $\newline$ $a = 10$
Calculate $a$ : Co-vertices are $(-2,0)$ and $(2,0)$ , so the minor axis is horizontal. The distance from the center to a co-vertex is the value of $b$ . Calculate $b$ using the distance formula or by observation. $\newline$ $b = 2$
Identify Minor Axis: The center $(h,k)$ is at the origin $(0,0)$ because the vertices and co-vertices are symmetrically arranged around the origin. $\newline$ $(h,k) = (0,0)$
Calculate 'b': Write the standard form equation of the ellipse using the values of 'a' and 'b'. Since the major axis is vertical, 'a' is associated with the 'y' term and 'b' is associated with the 'x' term. $\newline$ The standard form is $(x-h)^2/b^2 + (y-k)^2/a^2 = 1$ $\newline$ Plug in the values of 'a', 'b', 'h', and 'k'. $\newline$ $(x-0)^2/2^2 + (y-0)^2/10^2 = 1$
Determine Center: Simplify the equation. $\frac{x^2}{4} + \frac{y^2}{100} = 1$

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