If the length of the semi-major axis of an ellipse is 68 and e = 1/2, then the area of the rectangle formed by joining the vertices of latusrecta of the ellipse is equal to

Options

A. 6436
B. 6434
C. 6934
D. 6936

Answer

To solve the problem, we need to find the area of the rectangle formed by joining the vertices of the latus rectum of the ellipse given the semi-major axis length and eccentricity.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Length of the semi-major axis \( a = 68 \)
- Eccentricity \( e = \frac{1}{2} \)

2. **Calculate the Semi-Minor Axis \( b \)**:
The relationship between \( a \), \( b \), and \( e \) is given by the formula:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Rearranging this gives:
\[ b^2 = a^2(1 - e^2) \]
Now substituting the values:
\[ e^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ b^2 = 68^2 \left(1 - \frac{1}{4}\right) = 68^2 \cdot \frac{3}{4} \]
\[ b^2 = 4624 \cdot \frac{3}{4} = 3468 \]

3. **Calculate the Length of the Latus Rectum**:
The length of the latus rectum \( L \) is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \( b^2 \) and \( a \):
\[ L = \frac{2 \cdot 3468}{68} = \frac{6936}{68} = 102 \]

4. **Calculate the Area of the Rectangle**:
The area \( A \) of the rectangle formed by the vertices of the latus rectum is given by:
\[ A = \text{Length} \times \text{Width} \]
Here, the length of the rectangle is \( 2a \) and the width is \( L \):
\[ A = 2a \cdot L = 2 \cdot 68 \cdot 102 \]
\[ A = 136 \cdot 102 = 13872 \]

### Final Answer:
The area of the rectangle formed by joining the vertices of the latus rectum of the ellipse is \( 13872 \) square units.