In the parabola `y^(2) = 4ax`, the length of the chord passing through the vertex and inclined to the x-axis at `(pi)/(4)` is

Options

A. `4 sqrt (2)` a units
B. ` 2 sqrt(2)` a units
C. ` sqrt(2)` a units
D. none of these

Answer

To find the length of the chord passing through the vertex of the parabola \( y^2 = 4ax \) and inclined to the x-axis at an angle of \( \frac{\pi}{4} \) (or 45 degrees), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Equation of the Parabola**:
The given parabola is \( y^2 = 4ax \).

2. **Determine the Slope of the Chord**:
Since the chord is inclined at an angle of \( \frac{\pi}{4} \), the slope \( m \) of the line can be calculated as:
\[ m = \tan\left(\frac{\pi}{4}\right) = 1 \]

3. **Equation of the Chord**:
The equation of the line (chord) passing through the vertex (origin) with slope 1 is:
\[ y = x \]

4. **Substitute the Line Equation into the Parabola**:
To find the points where this line intersects the parabola, substitute \( y = x \) into the parabola's equation:
\[ (x)^2 = 4a(x) \]
This simplifies to:
\[ x^2 - 4ax = 0 \]

5. **Factor the Equation**:
Factor out \( x \):
\[ x(x - 4a) = 0 \]
This gives us two solutions:
\[ x = 0 \quad \text{and} \quad x = 4a \]

6. **Find the Corresponding y-values**:
For \( x = 0 \):
\[ y = 0 \]
For \( x = 4a \):
\[ y = 4a \]
Thus, the points of intersection are \( (0, 0) \) and \( (4a, 4a) \).

7. **Calculate the Length of the Chord**:
The length \( L \) of the chord can be calculated using the distance formula:
\[ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the points \( (0, 0) \) and \( (4a, 4a) \):
\[ L = \sqrt{(4a - 0)^2 + (4a - 0)^2} = \sqrt{(4a)^2 + (4a)^2} = \sqrt{16a^2 + 16a^2} = \sqrt{32a^2} = 4a\sqrt{2} \]

### Final Answer:
The length of the chord passing through the vertex and inclined to the x-axis at \( \frac{\pi}{4} \) is:
\[ \boxed{4a\sqrt{2}} \]