The equation of second degree `x^2+2sqrt2xy+2y^2+4x+4sqrt2y+1=0` represents a pair of straight lines.The distance between them is

Options

A. 2
B. `2sqrt(3)`
C. 4
D. `4sqrt(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 the equation of a pair of straight lines is given by:
\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]
From the given equation, we can identify the coefficients:
- \(A = 1\)
- \(H = \sqrt{2}\)
- \(B = 2\)
- \(G = 2\)
- \(F = 2\sqrt{2}\)
- \(C = 1\)

### Step 2: Use the formula for distance between the lines
The formula for the distance \(d\) between the two lines represented by the equation is:
\[ d = \frac{2\sqrt{G^2 - AB}}{A + B} \]

### Step 3: Calculate \(G^2\) and \(AB\)
Now we calculate \(G^2\) and \(AB\):
- \(G^2 = 2^2 = 4\)
- \(AB = 1 \cdot 2 = 2\)

### Step 4: Substitute values into the distance formula
Now substitute \(G^2\) and \(AB\) into the distance formula:
\[ d = \frac{2\sqrt{4 - 2}}{1 + 2} \]
\[ d = \frac{2\sqrt{2}}{3} \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ d = \frac{2\sqrt{2}}{3} \]

### Step 6: Final answer
Thus, the distance between the two lines is:
\[ d = \frac{2\sqrt{2}}{3} \]