A focus of an ellipse is at the origin. The directrix is the line `x =4` and the eccentricity is `1/2` Then the length of the semi-major axis is

Options

A. `8/3`
B. `2/3`
C. `4/3`
D. `5/3`

Answer

To solve the problem step by step, we need to use the properties of an ellipse and the relationships between its focus, directrix, and eccentricity.

### Step 1: Understand the given information
We are given:
- A focus of the ellipse at the origin (0, 0).
- The directrix is the line \( x = 4 \).
- The eccentricity \( e = \frac{1}{2} \).

### Step 2: Set up the relationship between the semi-major axis \( a \), eccentricity \( e \), and the directrix
The distance from the center of the ellipse to the directrix is given by the formula:
\[ OD = \frac{a}{e} \]
where \( O \) is the center of the ellipse, \( D \) is the directrix, and \( a \) is the semi-major axis.

### Step 3: Calculate the distance from the center to the directrix
Since the directrix is at \( x = 4 \) and the focus (which is also the center in this case) is at the origin (0, 0), the distance \( OD \) is:
\[ OD = 4 \]

### Step 4: Set up the equation using the eccentricity
From the relationship established earlier:
\[ OD = \frac{a}{e} \]
Substituting the known values:
\[ 4 = \frac{a}{\frac{1}{2}} \]

### Step 5: Solve for \( a \)
To find \( a \), we can rearrange the equation:
\[ 4 = 2a \quad \text{(since } \frac{1}{\frac{1}{2}} = 2\text{)} \]
Now, divide both sides by 2:
\[ a = \frac{4}{2} = 2 \]

### Step 6: Find the length of the semi-major axis
Now that we have \( a \), we can find the length of the semi-major axis:
\[ \text{Length of semi-major axis} = a = 2 \]

### Conclusion
The length of the semi-major axis of the ellipse is \( 2 \).

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