Question

If the distance between the foci of an ellipse is 6 and the length of the minor axis is 8, then the eccentricity is

• A. \(\frac{1}{\sqrt{5}}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{3}{5}\)
• D. \(\frac{4}{5}\)

Hint:

\(b^2 = a^2(1 - e^2)\) and \(2ae\) = distance between foci.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For ellipse, distance between foci = \(2ae\), minor axis = \(2b\).
Step 2: Detailed Explanation:
\(2ae = 6 \rightarrow ae = 3\)
\(2b = 8 \rightarrow b = 4\)
Also \(b^2 = a^2(1 - e^2) \rightarrow 16 = a^2 - a^2 e^2 = a^2 - 9\)
\(a^2 = 25 \rightarrow a = 5\)
\(e = \frac{3}{a} = \frac{3}{5}\)
Step 3: Final Answer:
\(\frac{3}{5}\).