Let (x,y) be any point on the parabola `y^2 = 4x`. Let P be the point that divides the line segment from (0,0) and (x,y) n the ratio 1:3. Then the locus of P is :

Options

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

Explanation

To find the locus of point \( P \) that divides the line segment from \( (0,0) \) to \( (x,y) \) in the ratio \( 1:3 \), where \( (x,y) \) lies on the parabola \( y^2 = 4x \), we can follow these steps:

### Step 1: Identify the coordinates of point \( P \)
Let the coordinates of point \( P \) be \( (h, k) \). Since \( P \) divides the segment in the ratio \( 1:3 \), we can use the section formula to find \( h \) and \( k \).

### Step 2: Apply the section formula
Using the section formula, the coordinates of point \( P \) can be calculated as follows:

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