The length of the chord of the parabola `x^(2) = 4y` having equations `x - sqrt(2) y + 4 sqrt(2) = 0` is

Options

A. `8 sqrt(2)`
B. `2 sqrt(11)`
C. `3 sqrt(2)`
D. `6 sqrt(3)`

Explanation

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

### Step 1: Rewrite the line equation
First, we rewrite the equation of the line in terms of \( y \):
\[ x - \sqrt{2}y + 4\sqrt{2} = 0 \implies \sqrt{2}y = x + 4\sqrt{2} \implies y = \frac{x}{\sqrt{2}} + 4 \]

### Step 2: Substitute \( y \) in the parabola equation
Next, we substitute this expression for \( y \) into the parabola equation \( x^2 = 4y \):
\[ x^2 = 4\left(\frac{x}{\sqrt{2}} + 4\right) \]
\[ x^2 = \frac{4x}{\sqrt{2}} + 16 \]

### Step 3: Clear the fraction
To eliminate the fraction, we multiply through by \( \sqrt{2} \):
\[ \sqrt{2}x^2 = 4x + 16\sqrt{2} \]

### Step 4: Rearrange into standard quadratic form
Rearranging gives us:
\[ \sqrt{2}x^2 - 4x - 16\sqrt{2} = 0 \]

### Step 5: Identify coefficients for the quadratic formula
From this quadratic equation, we can identify:
- \( a = \sqrt{2} \)
- \( b = -4 \)
- \( c = -16\sqrt{2} \)

### Step 6: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( x_1 + x_2 = -\frac{b}{a} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \)
- The product of the roots \( x_1 x_2 = \frac{c}{a} = \frac{-16\sqrt{2}}{\sqrt{2}} = -16 \)

### Step 7: Find \( y_1 \) and \( y_2 \)
Now, we need to find the corresponding \( y \) values for \( x_1 \) and \( x_2 \):
\[ y_1 = \frac{x_1}{\sqrt{2}} + 4, \quad y_2 = \frac{x_2}{\sqrt{2}} + 4 \]

### Step 8: Calculate the length of the chord
The length of the chord can be calculated using the distance formula:
\[ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Using the identities:
- \( (x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2 \)
- \( (y_2 - y_1)^2 = (y_1 + y_2)^2 - 4y_1 y_2 \)

We can substitute:
\[ L^2 = (2\sqrt{2})^2 - 4(-16) + \left(\left(\frac{2\sqrt{2}}{\sqrt{2}} + 8\right)^2 - 4(16)\right) \]
Calculating gives:
\[ L^2 = 8 + 64 + (10^2 - 64) = 8 + 64 + 36 = 108 \]
Thus, \( L = \sqrt{108} = 6\sqrt{3} \).

### Final Answer
The length of the chord is \( 6\sqrt{3} \).

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