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) to (x, y) in the ratio 1:3. Then locus of P is

Options

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

Answer

To find the locus of the 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 \( P \)
The coordinates of point \( P \) that divides the segment from \( (0, 0) \) to \( (x, y) \) in the ratio \( 1:3 \) can be found using the section formula. The section formula states that if a point divides a line segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of the dividing point \( (h, k) \) are given by:

\[ h = \frac{mx_2 + nx_1}{m+n}, \quad k = \frac{my_2 + ny_1}{m+n} \]

Here, \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (x, y) \), with \( m = 1 \) and \( n = 3 \).

### Step 2: Substitute the values into the section formula
Substituting the values into the section formula, we have:

\[ h = \frac{1 \cdot x + 3 \cdot 0}{1 + 3} = \frac{x}{4} \]

\[ k = \frac{1 \cdot y + 3 \cdot 0}{1 + 3} = \frac{y}{4} \]

Thus, the coordinates of point \( P \) are:

\[ P\left( \frac{x}{4}, \frac{y}{4} \right) \]

### Step 3: Express \( y \) in terms of \( x \)
Since \( (x, y) \) lies on the parabola \( y^2 = 4x \), we can express \( y \) in terms of \( x \):

\[ y = \sqrt{4x} = 2\sqrt{x} \]

### Step 4: Substitute \( y \) into the coordinates of \( P \)
Now, substitute \( y = 2\sqrt{x} \) into the coordinates of \( P \):

\[ P\left( \frac{x}{4}, \frac{2\sqrt{x}}{4} \right) = P\left( \frac{x}{4}, \frac{\sqrt{x}}{2} \right) \]

### Step 5: Relate \( h \) and \( k \)
Let \( h = \frac{x}{4} \) and \( k = \frac{\sqrt{x}}{2} \). We can express \( x \) in terms of \( h \):

\[ x = 4h \]

Now substitute \( x \) into the expression for \( k \):

\[ k = \frac{\sqrt{4h}}{2} = \frac{2\sqrt{h}}{2} = \sqrt{h} \]

### Step 6: Find the relationship between \( k \) and \( h \)
Squaring both sides gives:

\[ k^2 = h \]

### Step 7: Write the locus equation
Thus, the locus of point \( P \) is given by the equation:

\[ h = k^2 \]

### Final Locus Equation
Since we used \( h \) and \( k \) as the coordinates of point \( P \), we can rewrite the locus as:

\[ y = x^2 \]