Questions: For the ellipse shown, give the domain, range, center, vertices, and foci. x^2/9 + y^2/81 = 1 Find the domain. (Type your answer in interval notation.)

For the ellipse shown, give the domain, range, center, vertices, and foci.

x^2/9 + y^2/81 = 1

Find the domain.
(Type your answer in interval notation.)
Transcript text: For the ellipse shown, give the domain, range, center, vertices, and foci. \[ \frac{x^{2}}{9}+\frac{y^{2}}{81}=1 \] Find the domain. $\square$ (Type your answer in interval notation.)
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Solution

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Solution Steps

Step 1: Analyze the equation

The given equation of the ellipse is $\frac{x^2}{9} + \frac{y^2}{81} = 1$. This is a vertically oriented ellipse centered at the origin (0,0). The denominator under $x^2$ is $a^2 = 9$, so $a = 3$. The denominator under $y^2$ is $b^2 = 81$, so $b = 9$.

Step 2: Determine the vertices

The vertices are located at $(0, \pm b)$, so in this case, they are at $(0, 9)$ and $(0, -9)$.

Step 3: Find the domain

The domain is the set of all possible x-values. Since the ellipse is centered at the origin and extends horizontally by $a=3$ units in each direction, the domain is from -3 to 3, inclusive.

Final Answer: The domain is [-3, 3].

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