The length of the latus rectum of the parabola whose focus is a. `((u^2)/(2g)sin2alpha,-(u^2)/(2g)cos2alpha)` and directrix is `y=(u^2)/(2g)` is (a) `(u^2)/gcos^2alpha` (b) `(u^2)/gcos^2 2alpha` `(2u^2)/gcos^2 2alpha` (d) `(2u^2)/gcos^2alpha`

Options

A. `(u^(2))/(g)cos^(2)alpha`
B. `(u^(2))/(g)cos2alpha`
C. `(2u^(2))/(g)cos2alpha`
D. `(2u^(2))/(g)cos^(2)alpha`

Explanation

To solve the problem, we need to find the length of the latus rectum of the parabola given its focus and directrix.

### Step-by-Step Solution:

1. **Identify the Focus and Directrix:**
- The focus of the parabola is given as \( F = \left( \frac{u^2}{2g} \sin 2\alpha, -\frac{u^2}{2g} \cos 2\alpha \right) \).
- The directrix is given as \( y = \frac{u^2}{2g} \).

2. **Determine the Distance from the Focus to the Directrix:**
- The distance \( d \) from the focus to the directrix is the vertical distance since the directrix is a horizontal line.
- The y-coordinate of the focus is \( -\frac{u^2}{2g} \) and the y-coordinate of the directrix is \( \frac{u^2}{2g} \).
- Therefore, the distance \( d \) is:
\[ d = \left| -\frac{u^2}{2g} - \frac{u^2}{2g} \right| = \left| -\frac{u^2}{g} \right| = \frac{u^2}{g} \]

3. **Relate the Distance to the Parameter \( a \):**
- For a parabola, the distance from the focus to the directrix is \( 2a \). Thus, we have:
\[ 2a = \frac{u^2}{g} \]
- From this, we can solve for \( a \):
\[ a = \frac{u^2}{2g} \]

4. **Calculate the Length of the Latus Rectum:**
- The length of the latus rectum \( L \) of a parabola is given by the formula \( L = 4a \).
- Substituting the value of \( a \):
\[ L = 4 \left( \frac{u^2}{2g} \right) = \frac{4u^2}{2g} = \frac{2u^2}{g} \]

5. **Final Answer:**
- The length of the latus rectum is:
\[ \frac{2u^2}{g} \]
- Thus, the correct option is (c) \( \frac{2u^2}{g \cos^2 2\alpha} \).