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

Options

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

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step-by-Step Solution:

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

2. **Understand the Length of the Latus Rectum**:
The length of the latus rectum (LR) of a parabola in the form \(y^2 = 4ax\) is given by:
\[ \text{Length of Latus Rectum} = 4a \]

3. **Substitute the Point into the Parabola's Equation**:
We know that the parabola passes through the point (3, 2). Therefore, we will substitute \(x = 3\) and \(y = 2\) into the equation \(y^2 = 4ax\):
\[ 2^2 = 4a(3) \]

4. **Calculate \(4a\)**:
Simplifying the equation:
\[ 4 = 12a \]
To find \(a\), we divide both sides by 12:
\[ a = \frac{4}{12} = \frac{1}{3} \]

5. **Find \(4a\)**:
Now, we can find \(4a\):
\[ 4a = 4 \times \frac{1}{3} = \frac{4}{3} \]

6. **Conclusion**:
Therefore, the length of the latus rectum is:
\[ \frac{4}{3} \]

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

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