If the parabola `y^(2)=4ax` passes through (3, 2). Then the length of its latusrectum, is

Options

A. `2/3`
B. `4/3`
C. `1/3`
D. `4`

Explanation

To solve the problem step by step, we need to find the value of \( a \) for the parabola \( y^2 = 4ax \) that passes through the point (3, 2), and then determine the length of its latus rectum.

### Step-by-Step Solution:

1. **Write the equation of the parabola**:
The given equation of the parabola is:
\[ y^2 = 4ax \]

2. **Substitute the point (3, 2) into the equation**:
Since the parabola passes through the point (3, 2), we substitute \( x = 3 \) and \( y = 2 \) into the equation:
\[ 2^2 = 4a \cdot 3 \]

3. **Simplify the equation**:
Calculate \( 2^2 \):
\[ 4 = 4a \cdot 3 \]
This simplifies to:
\[ 4 = 12a \]

4. **Solve for \( a \)**:
To find \( a \), divide both sides by 12:
\[ a = \frac{4}{12} = \frac{1}{3} \]

5. **Write the equation of the parabola with the found value of \( a \)**:
Substitute \( a \) back into the parabola equation:
\[ y^2 = 4 \left(\frac{1}{3}\right)x \]
This simplifies to:
\[ y^2 = \frac{4}{3}x \]

6. **Find the length of the latus rectum**:
The length of the latus rectum of a parabola \( y^2 = 4ax \) is given by the formula:
\[ \text{Length of latus rectum} = 4a \]
Substitute the value of \( a \):
\[ \text{Length of latus rectum} = 4 \cdot \frac{1}{3} = \frac{4}{3} \]

### Final Answer:
The length of the latus rectum is:
\[ \frac{4}{3} \text{ units} \]