Step 1: Understanding the Question:
A parabola is defined as the set of points equidistant from a fixed point (focus) and a fixed line (directrix).
The standard equation \( y^2 = 4ax \) represents a parabola opening to the right with its vertex at the origin.
The "length of the latus rectum" is a standard property of parabolas and is numerically equal to the coefficient \( 4a \) in the standard equation.
Our goal is to find this value \( 4a \) by using a given point on the curve.
Step 2: Key Formula or Approach:
1. Substitute the coordinates of the point \( (x, y) = (3, 2) \) into the parabola equation \( y^2 = 4ax \).
2. Solve the resulting linear equation for the parameter \( a \).
3. Calculate the length of the latus rectum, which is defined as \( LR = 4a \).
Step 3: Detailed Explanation:
The point \( (3, 2) \) lies on the parabola \( y^2 = 4ax \).
This means when we set \( x = 3 \) and \( y = 2 \), the equation must hold true.
Substituting these values into the equation:
\[ (2)^2 = 4a(3) \]
\[ 4 = 12a \]
We can solve for \( a \) by dividing both sides by 12:
\[ a = \frac{4}{12} = \frac{1}{3} \]
The question asks for the length of the latus rectum.
By definition, for the parabola \( y^2 = 4ax \), the length of the latus rectum is the absolute value of the coefficient \( 4a \).
\[ \text{Length of Latus Rectum} = 4 \times \left(\frac{1}{3}\right) = \frac{4}{3} \]
Thus, the constant parameter \( a \) is \( 1/3 \) and the characteristic length of the chord passing through the focus is \( 4/3 \).
Step 4: Final Answer:
The length of the latus rectum is 4/3.