Locus of trisection point of any arbitrary double ordinate of the parabola `x^2 = 4by`, is -

Options

A. `9x^2=by`
B. `3x^2=2by`
C. `9x^2=4by`
D. `9x^2=2by`

Answer

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

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

### Step 2: Define the Double Ordinate
A double ordinate of a parabola is a line segment perpendicular to the axis of symmetry (the y-axis for this parabola) that intersects the parabola at two points. Let’s denote the points of intersection as \( A \) and \( B \).

### Step 3: Parametric Equations
Using the parametric equations for the parabola, we can express the coordinates of points \( A \) and \( B \):
- For point \( A \): \( A(-2bt, bt^2) \)
- For point \( B \): \( B(2bt, bt^2) \)

### Step 4: Find the Trisection Points
The trisection points divide the segment \( AB \) into three equal parts. We will find the coordinates of the trisection point \( P \) which divides \( AB \) in the ratio \( 2:1 \).

Using the section formula, the coordinates of point \( P \) can be calculated as follows:

#### X-coordinate of P:
\[ h = \frac{2 \cdot (2bt) + 1 \cdot (-2bt)}{2 + 1} = \frac{4bt - 2bt}{3} = \frac{2bt}{3} \]

#### Y-coordinate of P:
\[ k = \frac{2 \cdot (bt^2) + 1 \cdot (bt^2)}{2 + 1} = \frac{2bt^2 + bt^2}{3} = bt^2 \]

### Step 5: Substitute for \( t \)
From the expression for \( h \):
\[ h = \frac{2bt}{3} \implies t = \frac{3h}{2b} \]

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

### Step 7: Express in Terms of \( x \) and \( y \)
Since \( h = x \) and \( k = y \):
\[ y = \frac{9x^2}{4b} \]

### Step 8: Rearranging the Equation
Rearranging gives us:
\[ 9x^2 = 4by \]

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