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

Answer: (C)

$\frac{2}{\sqrt{5}}$

Answer 2

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:

$4 = 5e^2$

$e^2 = \frac{4}{5}$

$e = \frac{2}{\sqrt{5}}$