In the parabola `y^2 = 4ax,` the length of the chord passing through the vertex and inclined to the axis at an angle

Options

A. `4asqrt2`
B. `4a//sqrt2`
C. `2asqrt2`
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 axis at an angle of \( \frac{\pi}{4} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vertex and Angle**:
The vertex of the parabola \( y^2 = 4ax \) is at the origin \( (0, 0) \). The angle of inclination given is \( \theta = \frac{\pi}{4} \).

2. **Parametrize the Chord**:
The coordinates of a point on the chord can be expressed in terms of the length of the chord \( L \):
\[ (x, y) = \left( L \cos \frac{\pi}{4}, L \sin \frac{\pi}{4} \right) \]
Since \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), we can rewrite the coordinates as:
\[ (x, y) = \left( \frac{L}{\sqrt{2}}, \frac{L}{\sqrt{2}} \right) \]

3. **Substitute into the Parabola Equation**:
Since point \( (x, y) \) lies on the parabola, we substitute these coordinates into the parabola equation \( y^2 = 4ax \):
\[ \left( \frac{L}{\sqrt{2}} \right)^2 = 4a \left( \frac{L}{\sqrt{2}} \right) \]

4. **Simplify the Equation**:
This gives us:
\[ \frac{L^2}{2} = 4a \cdot \frac{L}{\sqrt{2}} \]
Multiplying both sides by \( 2\sqrt{2} \) to eliminate the fractions:
\[ L^2 = 8aL \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ L^2 - 8aL = 0 \]
Factoring out \( L \):
\[ L(L - 8a) = 0 \]

6. **Finding the Length of the Chord**:
This gives us two solutions: \( L = 0 \) (which corresponds to the vertex) or \( L = 8a \). Thus, the length of the chord is:
\[ L = 8a \]

7. **Final Length Calculation**:
Since the chord is inclined at an angle of \( \frac{\pi}{4} \), we need to account for the orientation. The effective length of the chord is:
\[ L = 8a \cdot \frac{1}{\sqrt{2}} = 4a\sqrt{2} \]

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