The length of the chord of the parabola `y^(2)=4ax` whose equation is
`y-x sqrt2+4asqrt2=0` is

Options

A. `2sqrt11a`
B. `4sqrt2a`
C. `8sqrt2a`
D. `6sqrt3a`

Answer

To find the length of the chord of the parabola given by the equation \( y^2 = 4ax \) and the line given by \( y - x\sqrt{2} + 4a\sqrt{2} = 0 \), we can follow these steps:

### Step 1: Identify the slope and intercept of the line
The equation of the line can be rewritten in slope-intercept form:
\[ y = x\sqrt{2} - 4a\sqrt{2} \]
From this, we can see that the slope \( m \) is \( \sqrt{2} \) and the y-intercept \( c \) is \( -4a\sqrt{2} \).

**Hint:** To find the slope and intercept of a line, rearrange the equation into the form \( y = mx + c \).

### Step 2: Use the formula for the length of the chord
The formula for the length \( L \) of the chord of a parabola \( y^2 = 4ax \) for a line \( y = mx + c \) is given by:
\[ L = \frac{4}{m^2} \sqrt{a(1 + m^2)(a - mc)} \]

**Hint:** Remember to substitute the values of \( m \) and \( c \) correctly into the chord length formula.

### Step 3: Substitute the values into the formula
Substituting \( m = \sqrt{2} \) and \( c = -4a\sqrt{2} \) into the formula:
- Calculate \( m^2 \):
\[ m^2 = (\sqrt{2})^2 = 2 \]
- Substitute into the formula:
\[ L = \frac{4}{2} \sqrt{a(1 + 2)(a - \sqrt{2}(-4a\sqrt{2}))} \]
This simplifies to:
\[ L = 2 \sqrt{a(3)(a + 8a)} = 2 \sqrt{a(3)(9a)} = 2 \sqrt{27a^2} \]

**Hint:** Simplifying the expression inside the square root can help you find the final length.

### Step 4: Simplify the expression
Now simplify \( 2 \sqrt{27a^2} \):
\[ L = 2 \cdot 3a\sqrt{3} = 6a\sqrt{3} \]

**Hint:** Factor out constants and simplify square roots to get the final answer.

### Final Answer
Thus, the length of the chord is:
\[ \boxed{6a\sqrt{3}} \]