Question

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

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

Answer 1

The Correct Option is C

Solution: Let a be the semi-major axis, b the semi-minor axis, and 2c the distance between the foci of the ellipse. The eccentricity e is defined as \( e = \frac{c}{a} \).

Since the length of the minor axis is equal to half of the distance between the foci, we have:

\[ 2b = \frac{1}{2} \times 2c \Rightarrow 2b = c \]

Substitute \( c = ae \) into the equation:

\[ 2b = ae \]

Using the relationship \( b = a\sqrt{1 - e^2} \), we substitute for b:

\[ 2a\sqrt{1 - e^2} = ae \]

Divide by a:

\[ 2\sqrt{1 - e^2} = e \]

Square both sides:

\[ 4(1 - e^2) = e^2 \]

Expanding and rearranging terms:

\[ 4 = 5e^2 \]

\[ e^2 = \frac{4}{5} \]

\[ e = \frac{2}{\sqrt{5}} \]