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:

In an ellipse, $a^{2}e^{2} = a^{2} - b^{2}$.

Answer 1

The Correct Option is C

Step 1: Given Values
Distance between foci $2ae = 6 \Rightarrow ae = 3$. Length of minor axis $2b = 8 \Rightarrow b = 4 \Rightarrow b^{2} = 16$.
Step 2: Use Ellipse Identity
$b^{2} = a^{2}(1-e^{2}) \Rightarrow 16 = a^{2} - (ae)^{2}$. $16 = a^{2} - 3^{2} \Rightarrow a^{2} = 16 + 9 = 25 \Rightarrow a = 5$.
Step 3: Calculate Eccentricity
$e = \frac{ae}{a} = \frac{3}{5}$.
Final Answer: (c)