PQ is a double ordinate of a parabola `y^(2)=4ax`. The locus of its points of trisection is another parabola length of whose latus rectum is k times the length of the latus rectum of the given parabola, the value of k is

Options

A. `1//9`
B. `1//3`
C. `2//3`
D. none of these

Answer

To solve the problem, we need to find the value of \( k \) based on the given conditions regarding the parabola \( y^2 = 4ax \) and the double ordinate \( PQ \).

### Step-by-Step Solution:

1. **Identify the Given Parabola**:
The equation of the parabola is given as:
\[ y^2 = 4ax \]
The length of the latus rectum \( L_1 \) of this parabola is given by:
\[ L_1 = 4a \]

2. **Define the Double Ordinate**:
A double ordinate \( PQ \) is a line segment parallel to the y-axis. Let the x-coordinate of the double ordinate be \( x = a \) and the y-coordinates be \( y = h \) and \( y = -h \). Thus, the points \( P \) and \( Q \) can be represented as:
\[ P(a, h) \quad \text{and} \quad Q(a, -h) \]

3. **Find the Points of Trisection**:
The points of trisection of the segment \( PQ \) divide it into three equal parts. The coordinates of the points of trisection \( M_1 \) and \( M_2 \) can be calculated as follows:
- The first point of trisection \( M_1 \) is:
\[ M_1 = \left(a, \frac{2h}{3}\right) \]
- The second point of trisection \( M_2 \) is:
\[ M_2 = \left(a, -\frac{2h}{3}\right) \]

4. **Determine the Locus of the Points of Trisection**:
To find the locus of the points \( M_1 \) and \( M_2 \), we express \( h \) in terms of \( x \) and \( y \):
- From the coordinates of \( M_1 \):
\[ y = \frac{2h}{3} \implies h = \frac{3y}{2} \]
- Substitute \( h \) into the parabola equation:
\[ y^2 = 4a\left(\frac{3y}{2}\right)^2 \]
Simplifying this gives:
\[ y^2 = 4a \cdot \frac{9y^2}{4} \implies y^2 = 9ay^2 \implies y^2 = 9ax \]

5. **Identify the New Parabola**:
The equation \( y^2 = 9ax \) represents another parabola. The length of the latus rectum \( L_2 \) of this new parabola is given by:
\[ L_2 = 4b = 4 \cdot \frac{9a}{4} = 9a \]

6. **Calculate the Ratio \( k \)**:
Now, we can find \( k \) by comparing the lengths of the latus rectum:
\[ k = \frac{L_2}{L_1} = \frac{9a}{4a} = \frac{9}{4} \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{\frac{9}{4}} \]