The locus of the middle points of the chords of the parabola `y^(2)=4ax`, which passes through the origin is :

Options

A. `y^2 = ax`
B. `y^2 = 2ax`
C. `y^2 = 4ax`
D. `x^2 = 4ay`

Answer

To find the locus of the midpoints of the chords of the parabola \( y^2 = 4ax \) that pass through the origin, we can follow these steps:

### Step 1: Understand the equation of the parabola
The given parabola is \( y^2 = 4ax \). This is a standard form of a parabola that opens to the right.

### Step 2: Equation of the chord
Any chord of the parabola that passes through the origin can be represented by the line equation \( y = mx \), where \( m \) is the slope of the line.

### Step 3: Find the points of intersection
To find the points where this line intersects the parabola, substitute \( y = mx \) into the parabola's equation:

\[ (mx)^2 = 4ax \]

This simplifies to:

\[ m^2x^2 = 4ax \]

Rearranging gives:

\[ m^2x^2 - 4ax = 0 \]

Factoring out \( x \):

\[ x(m^2x - 4a) = 0 \]

This gives us two solutions: \( x = 0 \) (the origin) and \( x = \frac{4a}{m^2} \).

### Step 4: Find the corresponding \( y \) values
For \( x = \frac{4a}{m^2} \), substitute back into the line equation to find \( y \):

\[ y = m \left(\frac{4a}{m^2}\right) = \frac{4a}{m} \]

### Step 5: Identify the midpoints of the chord
The points of intersection are \( (0, 0) \) and \( \left(\frac{4a}{m^2}, \frac{4a}{m}\right) \). The midpoint \( M \) of the chord can be calculated as:

\[ M = \left( \frac{0 + \frac{4a}{m^2}}{2}, \frac{0 + \frac{4a}{m}}{2} \right) = \left( \frac{2a}{m^2}, \frac{2a}{m} \right) \]

### Step 6: Eliminate \( m \) to find the locus
Now we have the coordinates of the midpoint \( M\) as \( \left( \frac{2a}{m^2}, \frac{2a}{m} \right) \). Let \( x = \frac{2a}{m^2} \) and \( y = \frac{2a}{m} \).

From \( x = \frac{2a}{m^2} \), we can express \( m^2 \):

\[ m^2 = \frac{2a}{x} \]

Substituting this into the expression for \( y \):

\[ y = \frac{2a}{m} = \frac{2a}{\sqrt{\frac{2a}{x}}} = \frac{2a \sqrt{x}}{\sqrt{2a}} = \sqrt{2a} \cdot \sqrt{x} \]

Squaring both sides gives:

\[ y^2 = 2ax \]

### Conclusion
Thus, the locus of the midpoints of the chords of the parabola \( y^2 = 4ax \) that pass through the origin is:

\[ \boxed{y^2 = 2ax} \]