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.)
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.)
Solution
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.