The locus of the trisection point of any arbitrary double ordinate of the parabola `x^(2)=4y`, is

Options

A. `9x^(2)=y`
B. `3x^(2)=2y`
C. `9x^(2)=4y`
D. `9x^(2)=2y`

Answer

To find the locus of the trisection point of any arbitrary double ordinate of the parabola \( x^2 = 4y \), we can follow these steps:

### Step 1: Understand the Parabola
The given parabola is \( x^2 = 4y \). This is a standard form of a parabola that opens upwards with its vertex at the origin (0,0).

### Step 2: Identify the Double Ordinate
A double ordinate of the parabola is a line segment parallel to the y-axis that intersects the parabola at two points. Let’s denote the points of intersection as \( A(2t, t^2) \) and \( B(-2t, t^2) \), where \( t \) is a parameter.

### Step 3: Find the Trisection Points
The trisection points divide the line segment \( AB \) into three equal parts. Let \( P \) be the trisection point that divides \( AB \) in the ratio \( 1:2 \). Using the section formula, we can find the coordinates of point \( P \).

The coordinates of \( P \) are given by:
\[ P\left( h, k \right) = \left( \frac{1 \cdot (-2t) + 2 \cdot (2t)}{1 + 2}, \frac{1 \cdot t^2 + 2 \cdot t^2}{1 + 2} \right) \]

### Step 4: Calculate the Coordinates of \( P \)
Substituting the values into the formula:
\[ h = \frac{-2t + 4t}{3} = \frac{2t}{3} \]
\[ k = \frac{t^2 + 2t^2}{3} = \frac{3t^2}{3} = t^2 \]

### Step 5: Express \( t \) in Terms of \( h \)
From \( h = \frac{2t}{3} \), we can express \( t \) as:
\[ t = \frac{3h}{2} \]

### Step 6: Substitute \( t \) into the Equation for \( k \)
Now substitute \( t \) into the equation for \( k \):
\[ k = t^2 = \left( \frac{3h}{2} \right)^2 = \frac{9h^2}{4} \]

### Step 7: Rearranging the Equation
Rearranging the equation gives us:
\[ 9h^2 = 4k \]

### Step 8: Replace \( h \) and \( k \) with \( x \) and \( y \)
Since \( h \) and \( k \) represent the coordinates of point \( P \), we can replace them with \( x \) and \( y \):
\[ 9x^2 = 4y \]

### Conclusion
Thus, the locus of the trisection point of any arbitrary double ordinate of the parabola \( x^2 = 4y \) is given by:
\[ 9x^2 = 4y \]