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`

Explanation

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 Parabola
The given parabola is \( y^2 = 4ax \). This is a standard form of a parabola that opens to the right.

### Step 2: Define the Midpoint of the Chord
Let the midpoint of the chord be denoted as \( (H, K) \).

### Step 3: Write the Equation of the Chord
The equation of the chord passing through the point \( (x_1, y_1) \) on the parabola can be expressed as:
\[ y - K = m(x - H) \]
where \( m \) is the slope of the chord.

### Step 4: Use the Condition of the Chord
Since the chord passes through the origin, we can substitute \( (0, 0) \) into the chord equation:
\[ 0 - K = m(0 - H) \]
This simplifies to:
\[ -K = -mH \quad \Rightarrow \quad K = mH \]

### Step 5: Substitute into the Parabola Equation
The points \( (x_1, y_1) \) on the parabola satisfy the equation \( y_1^2 = 4ax_1 \). The coordinates of the endpoints of the chord can be represented in terms of \( H \) and \( K \) as:
\[ y_1 = K + m(H - H_1) \quad \text{and} \quad x_1 = H + \frac{K}{m} \]

### Step 6: Relate the Midpoint to the Parabola
Using the midpoint \( (H, K) \) and the fact that the chord passes through the origin, we can derive:
\[ K^2 = 4aH \]
This is derived from the relationship of the points on the parabola.

### Step 7: Rearranging the Equation
Rearranging gives us:
\[ K^2 - 4aH = 0 \]
This can be rewritten as:
\[ K^2 = 4aH \]

### Step 8: Replace \( H \) and \( K \)
To find the locus, we replace \( H \) and \( K \) with \( x \) and \( y \):
\[ y^2 = 4ax \]

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