The equation of second degree
`x^(2)+2sqrt(2)xy+2y^(2)+4x+4sqrt(2)y+1=0`
represents a pair of straight lines, the distance between them is

Options

A. 4
B. `4//sqrt(3)`
C. `2`
D. `2//sqrt(3)`

Answer

To find the distance between the pair of straight lines represented by the equation

\[ x^2 + 2\sqrt{2}xy + 2y^2 + 4x + 4\sqrt{2}y + 1 = 0, \]

we will follow these steps:

### Step 1: Identify the coefficients
The general form of a second-degree equation is given by:

\[ ax^2 + 2gxy + by^2 + 2fx + 2hy + c = 0. \]

From the given equation, we can identify the coefficients:
- \( a = 1 \)
- \( g = \sqrt{2} \)
- \( b = 2 \)
- \( f = 2 \)
- \( h = 2\sqrt{2} \)
- \( c = 1 \)

### Step 2: Check if the lines are parallel
For the equation to represent a pair of parallel lines, the condition \( g^2 = ab \) must hold true.

Calculating \( g^2 \) and \( ab \):
- \( g^2 = (\sqrt{2})^2 = 2 \)
- \( ab = 1 \cdot 2 = 2 \)

Since \( g^2 = ab \), the lines are indeed parallel.

### Step 3: Calculate the distance between the parallel lines
The formula for the distance \( d \) between two parallel lines represented by the equation is given by:

\[ d = \frac{2\sqrt{g^2 - ac}}{\sqrt{a + b}} \]

Substituting the values we have:
- \( g^2 = 2 \)
- \( a = 1 \)
- \( c = 1 \)
- \( b = 2 \)

Calculating \( g^2 - ac \):
\[ g^2 - ac = 2 - (1 \cdot 1) = 2 - 1 = 1 \]

Now substituting into the distance formula:
\[ d = \frac{2\sqrt{1}}{\sqrt{1 + 2}} = \frac{2 \cdot 1}{\sqrt{3}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

### Step 4: Final result
Thus, the distance between the lines is:

\[ d = \frac{2\sqrt{3}}{3} \]