The difference between the lengths of the major axis and the latus rectum of an ellipse is (i) ae (ii) 2ae (iii) `ae^(2)` (iv) `2ae^(2)`

Options

A. ae
B. 2ae
C. `ae^(2)`
D. `2ae^(2)`

Answer

To solve the problem of finding the difference between the lengths of the major axis and the latus rectum of an ellipse, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths of the major axis and the latus rectum**:
- The length of the major axis of an ellipse is given by \(2a\).
- The length of the latus rectum is given by \(\frac{2b^2}{a}\).

2. **Write the expression for the difference**:
- Let \(d\) be the difference between the lengths of the major axis and the latus rectum.
- Therefore, we can express this as:
\[ d = 2a - \frac{2b^2}{a} \]

3. **Substitute \(b^2\) in terms of \(a\) and \(e\)**:
- We know that for an ellipse, \(b^2 = a^2(1 - e^2)\), where \(e\) is the eccentricity of the ellipse.
- Substitute \(b^2\) into the expression for \(d\):
\[ d = 2a - \frac{2(a^2(1 - e^2))}{a} \]

4. **Simplify the expression**:
- This simplifies to:
\[ d = 2a - 2a(1 - e^2) \]
- Distributing the terms gives:
\[ d = 2a - 2a + 2ae^2 \]
- Thus, we have:
\[ d = 2ae^2 \]

5. **Conclusion**:
- The difference between the lengths of the major axis and the latus rectum is \(2ae^2\).

### Final Answer:
The correct option is (iv) \(2ae^2\).

---