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:
- \( \frac{\sqrt{5}}{3} \)
- \( \frac{\sqrt{3}}{2} \)
- \( \frac{2}{\sqrt{5}} \)
- \( \frac{1}{\sqrt{3}} \)
The Correct Option is C
Approach Solution - 1
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:
\[ 4 = 5e^2 \]
\[ e^2 = \frac{4}{5} \]
\[ e = \frac{2}{\sqrt{5}} \]
Approach Solution -2
To determine the eccentricity of the ellipse, we start with the given condition: the length of the minor axis is equal to half of the distance between the foci.
An ellipse is defined by its semi-major axis \( a \) and semi-minor axis \( b \), with the relationship between the eccentricity \( e \), semi-major axis, and semi-minor axis given by:
\[ c^2 = a^2 - b^2 \]
\[ e = \frac{c}{a} \]
where \( c \) is the distance from the center to one focus.
The distance between the foci is \( 2c \), and according to the problem, the length of the minor axis \( 2b \) is equal to half of this distance:
\[ 2b = \frac{1}{2} \times 2c \]
\[ 2b = c \]
Thus, we have:
\[ b = \frac{c}{2} \]
Substituting \( b = \frac{c}{2} \) into the ellipse equation \( c^2 = a^2 - b^2 \), we get:
\[ c^2 = a^2 - \left(\frac{c}{2}\right)^2 \]
Simplifying, we find:
\[ c^2 = a^2 - \frac{c^2}{4} \]
\[ 4c^2 = 4a^2 - c^2 \]
\[ 5c^2 = 4a^2 \]
Hence, the expression for \( a^2 \) in terms of \( c^2 \) is:
\[ a^2 = \frac{5}{4}c^2 \]
Plugging this back into the equation for \( e \):
\[ e = \frac{c}{a} = \frac{c}{\sqrt{\frac{5}{4}}c} = \frac{2}{\sqrt{5}} \]
This matches with the given correct answer.
Thus, the eccentricity of the ellipse is \(\frac{2}{\sqrt{5}}\).
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