Step 1: Understanding the Question:
An ellipse is a closed curve where the sum of distances from any point on the curve to two fixed points (foci) is constant.
Foci located at \( (\pm 3, 0) \) indicate that the ellipse is centered at the origin and its major axis lies along the x-axis.
Our goal is to find the values of \( a^2 \) and \( b^2 \) for the standard equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) using the given focus coordinates and eccentricity.
Step 2: Key Formula or Approach:
1. Coordinates of foci for a horizontal ellipse: \( (\pm ae, 0) \).
2. Relation between axes and eccentricity: \( b^2 = a^2(1 - e^2) \).
3. Given: \( ae = 3 \) and \( e = 1/3 \).
Step 3: Detailed Explanation:
Step 3.1: Finding 'a':
We know that the distance of each focus from the center is \( ae \).
\[ ae = 3 \]
Substitute the given value of \( e = 1/3 \):
\[ a \times \frac{1}{3} = 3 \]
\[ a = 3 \times 3 = 9 \]
Thus, \( a^2 = 9^2 = 81 \).
Step 3.2: Finding 'b\(^2\)':
Use the identity \( b^2 = a^2(1 - e^2) \):
\[ b^2 = 81 \left(1 - \left(\frac{1}{3}\right)^2\right) \]
\[ b^2 = 81 \left(1 - \frac{1}{9}\right) \]
\[ b^2 = 81 \left(\frac{8}{9}\right) \]
\[ b^2 = 9 \times 8 = 72 \]
Step 3.3: Writing the final equation:
Substitute \( a^2 = 81 \) and \( b^2 = 72 \) into the standard form:
\[ \frac{x^2}{81} + \frac{y^2}{72} = 1 \]
Step 4: Final Answer:
The equation of the ellipse is \( \frac{x^2}{81} + \frac{y^2}{72} = 1 \).