If the parabola of `y^2=4ax` passes through the point (3,2), find the length of its latus rectum.

Answer

To solve the problem, we need to find the value of \( a \) for the parabola given by the equation \( y^2 = 4ax \) that passes through the point (3, 2). Then, we will use that value to find the length of the latus rectum.

### Step-by-Step Solution:

1. **Substituting the Point into the Parabola Equation:**
The parabola is given by the equation:
\[ y^2 = 4ax \]
We know that the parabola passes through the point (3, 2). Therefore, we substitute \( x = 3 \) and \( y = 2 \) into the equation:
\[ 2^2 = 4a(3) \]

2. **Calculating the Left Side:**
Calculate \( 2^2 \):
\[ 4 = 4a(3) \]

3. **Simplifying the Equation:**
Now, simplify the equation:
\[ 4 = 12a \]

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

5. **Finding 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 \( a = \frac{1}{3} \) into the formula:
\[ \text{Length of Latus Rectum} = 4 \times \frac{1}{3} = \frac{4}{3} \]

### Final Answer:
The length of the latus rectum is \( \frac{4}{3} \) units.

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