The distance between the foci of an ellipse is equal to half of its minor axis then eccentricity is

Options

A. `(1)/sqrt(2)`
B. `(1)/sqrt(5)`
C. `(2)/sqrt(3)`
D. `(1)/sqrt(3)`

Answer

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Ellipse**:
We start with the standard form of the ellipse given by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. We know that \( a > b \).

**Hint**: Remember that in an ellipse, the major axis is always longer than the minor axis.

2. **Distance Between Foci**:
The distance between the foci of the ellipse is given by:
\[ 2ae \]
where \( e \) is the eccentricity of the ellipse.

**Hint**: The foci are located at a distance of \( ae \) from the center along the major axis.

3. **Length of the Minor Axis**:
The length of the minor axis is:
\[ 2b \]
Therefore, half of the minor axis is:
\[ b \]

**Hint**: The minor axis is perpendicular to the major axis and is shorter in length.

4. **Setting Up the Equation**:
According to the problem, the distance between the foci is equal to half of the minor axis:
\[ 2ae = b \]

**Hint**: This relationship is key to finding the eccentricity.

5. **Rearranging the Equation**:
We can rearrange the equation to express \( b \) in terms of \( a \) and \( e \):
\[ e = \frac{b}{2a} \]

**Hint**: Isolate \( e \) to find a relationship between \( b \) and \( a \).

6. **Using the Eccentricity Formula**:
The eccentricity \( e \) can also be expressed using the relationship between \( a \) and \( b \):
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

**Hint**: This formula connects the eccentricity with the lengths of the axes.

7. **Substituting \( b \)**:
From our earlier equation, substitute \( b = 2ae \) into the eccentricity formula:
\[ e = \sqrt{1 - \frac{(2ae)^2}{a^2}} = \sqrt{1 - 4e^2} \]

**Hint**: Make sure to square the \( 2ae \) correctly.

8. **Squaring Both Sides**:
Now, square both sides to eliminate the square root:
\[ e^2 = 1 - 4e^2 \]

**Hint**: This step will help us form a quadratic equation.

9. **Rearranging the Equation**:
Rearranging gives:
\[ 5e^2 = 1 \]

**Hint**: Combine like terms to simplify the equation.

10. **Solving for \( e^2 \)**:
Divide both sides by 5:
\[ e^2 = \frac{1}{5} \]

**Hint**: This gives us the square of the eccentricity.

11. **Finding \( e \)**:
Finally, take the square root to find \( e \):
\[ e = \frac{1}{\sqrt{5}} \]

**Hint**: Remember that eccentricity is always a positive value.

### Final Answer:
The eccentricity \( e \) of the ellipse is:
\[ \frac{1}{\sqrt{5}} \]