Question

The equation of the ellipse with foci at ($\pm$3, 0) and the eccentricity as 1/3 is:

• A. $\frac{x^{2}}{81} + \frac{y^{2}}{72} = 1$
• B. $\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1$
• C. $\frac{x^{2}}{8} + \frac{y^{2}}{9} = 1$
• D. $\frac{x^{2}}{3} + \frac{y^{2}}{2} = 1$

Hint:

If foci are $(\pm c, 0)$, then $a$ must be greater than $c$. Here $a=9 > c=3$, confirming our steps.

Answer 1

The Correct Option is A

Step 1: Concept
For an ellipse centered at the origin with foci on the x-axis, foci are at $(\pm ae, 0)$ and $b^{2} = a^{2}(1 - e^{2})$.

Step 2: Meaning
Given $ae = 3$ and $e = 1/3$. Substitute $e$: $a(1/3) = 3 \implies a = 9$, so $a^{2} = 81$.

Step 3: Analysis
$b^{2} = 81(1 - (1/3)^{2}) = 81(1 - 1/9) = 81(8/9) = 72$.

Step 4: Conclusion
The equation is $x^{2}/81 + y^{2}/72 = 1$.
Final Answer: (A)