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

Options

A. `frac{2}{sqrt 5}`
B. `frac{3}{sqrt 5}`
C. `frac{2}{sqrt 7}`
D. `frac{3}{sqrt 7}`

Answer

To solve the problem, we need to find the eccentricity of the ellipse given that the length of the minor axis is equal to half of the distance between the foci.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The length of the minor axis of an ellipse is given by \(2b\), where \(b\) is the semi-minor axis.
- The distance between the foci of an ellipse is given by \(2c\), where \(c = ae\) (with \(a\) being the semi-major axis and \(e\) being the eccentricity).

2. **Set Up the Equation**:
- According to the problem, the length of the minor axis is equal to half of the distance between the foci:
\[ 2b = \frac{1}{2}(2c) \]
- This simplifies to:
\[ 2b = c \]

3. **Substitute for \(c\)**:
- Since \(c = ae\), we can substitute \(c\) in the equation:
\[ 2b = ae \]

4. **Express \(b\) in terms of \(a\) and \(e\)**:
- Rearranging gives:
\[ b = \frac{ae}{2} \]

5. **Use the Relationship Between \(a\), \(b\), and \(e\)**:
- We know the relationship for ellipses:
\[ b^2 = a^2(1 - e^2) \]
- Substitute \(b = \frac{ae}{2}\) into this equation:
\[ \left(\frac{ae}{2}\right)^2 = a^2(1 - e^2) \]
- This simplifies to:
\[ \frac{a^2e^2}{4} = a^2(1 - e^2) \]

6. **Cancel \(a^2\) (assuming \(a \neq 0\))**:
- Dividing both sides by \(a^2\) gives:
\[ \frac{e^2}{4} = 1 - e^2 \]

7. **Solve for \(e^2\)**:
- Rearranging the equation:
\[ e^2 + \frac{e^2}{4} = 1 \]
- This can be written as:
\[ \frac{4e^2 + e^2}{4} = 1 \implies \frac{5e^2}{4} = 1 \]
- Multiplying both sides by 4:
\[ 5e^2 = 4 \]
- Dividing by 5:
\[ e^2 = \frac{4}{5} \]

8. **Find \(e\)**:
- Taking the square root:
\[ e = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

### Final Answer:
The eccentricity \(e\) of the ellipse is \( \frac{2}{\sqrt{5}} \).

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