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Write the equation in standard form for the ellipse with vertices $(-9,0)$ and $(9,0)$ , and co-vertices $(0,5)$ and $(0,-5)$ .

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Write equations of ellipses in standard form using properties

Full solution

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

(-9,0)

and

(9,0)

, and co-vertices

(0,5)

and

(0,-5)

.

Calculate semi-major axis: Vertices are $(-9,0)$ and $(9,0)$ , so the major axis is horizontal and the length of the major axis is $2a$ , where $a$ is the semi-major axis. $\newline$ Calculate $a$ : $a = \text{distance from center to vertex} = 9 - 0 = 9$ .
Calculate semi-minor axis: Co-vertices are $(0,5)$ and $(0,-5)$ , so the minor axis is vertical and the length of the minor axis is $2b$ , where $b$ is the semi-minor axis. $\newline$ Calculate $b$ : $b = \text{distance from center to co-vertex} = 5 - 0 = 5$ .
Find center coordinates: The center $(h,k)$ is at the midpoint of the vertices, which is $(0,0)$ since the vertices are equidistant from the origin.
Standard form of ellipse equation: The standard form of the equation of an ellipse with a horizontal major axis is $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$ . Plug in $h=0$ , $k=0$ , $a=9$ , and $b=5$ into the equation.
Simplify the equation: The equation becomes $(x-0)^2/9^2 + (y-0)^2/5^2 = 1$ . $\newline$ Simplify the equation: $x^2/81 + y^2/25 = 1$ .

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