Let O be the vertex and Q be any point on the parabola,`x^2=""8y` . It the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is : (1) `x^2=""y` (2) `y^2=""x` (3) `y^2=""2x` (4) `x^2=""2y`

Options

A. `x^2=y`
B. `y^2=x`
C. `y^2=2x`
D. `x^2=2y`

Answer

To find the locus of point P that divides the line segment OQ internally in the ratio 1:3, where O is the vertex of the parabola \( x^2 = 8y \) and Q is any point on the parabola, we can follow these steps:

### Step 1: Identify the coordinates of point Q on the parabola
The equation of the parabola is given as \( x^2 = 8y \). The general coordinates of any point Q on this parabola can be expressed in terms of a parameter \( t \):
\[ Q(4t, 2t^2) \]
Here, \( x = 4t \) and \( y = 2t^2 \).

### Step 2: Identify the coordinates of point O
The vertex O of the parabola is at the origin, which has coordinates:
\[ O(0, 0) \]

### Step 3: Use the section formula to find the coordinates of point P
Point P divides the line segment OQ in the ratio 1:3. According to the section formula, if a point divides the line segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m_1:m_2 \), then the coordinates of the point can be calculated as:
\[ P\left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right) \]
In our case, \( m_1 = 1 \), \( m_2 = 3 \), \( (x_1, y_1) = (0, 0) \) (point O), and \( (x_2, y_2) = (4t, 2t^2) \) (point Q).

### Step 4: Calculate the x-coordinate of point P
Using the section formula for the x-coordinate:
\[ H = \frac{1 \cdot 4t + 3 \cdot 0}{1 + 3} = \frac{4t}{4} = t \]

### Step 5: Calculate the y-coordinate of point P
Using the section formula for the y-coordinate:
\[ K = \frac{1 \cdot 2t^2 + 3 \cdot 0}{1 + 3} = \frac{2t^2}{4} = \frac{t^2}{2} \]

### Step 6: Express K in terms of H
From the previous steps, we have:
\[ H = t \quad \text{and} \quad K = \frac{t^2}{2} \]
We can express \( K \) in terms of \( H \):
\[ K = \frac{H^2}{2} \]

### Step 7: Find the locus of point P
To find the locus, we can rearrange the equation:
\[ 2K = H^2 \]
This can be rewritten as:
\[ H^2 = 2K \]
Substituting back \( H \) and \( K \) with \( x \) and \( y \) respectively, we get:
\[ x^2 = 2y \]

### Conclusion
Thus, the locus of point P is given by the equation:
\[ x^2 = 2y \]
This corresponds to option (4).