The equation `8x^2 +8xy +2y^2+26x +13y+15=0` represents a pair of straight lines. The distance between them is

Options

A. `(7)/(sqrt5)`
B. `(7)/(2sqrt5)`
C. `sqrt((7)/(5))`
D. none of these

Explanation

To find the distance between the pair of straight lines represented by the equation \(8x^2 + 8xy + 2y^2 + 26x + 13y + 15 = 0\), we can follow these steps:

### Step 1: Identify coefficients
The given equation can be compared to the general form of a pair of straight lines, which is:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the equation \(8x^2 + 8xy + 2y^2 + 26x + 13y + 15 = 0\), we can identify the coefficients:
- \(a = 8\)
- \(b = 2\)
- \(h = 4\) (since \(2h = 8\))
- \(g = 13\) (since \(2g = 26\))
- \(f = \frac{13}{2}\) (since \(2f = 13\))
- \(c = 15\)

### Step 2: Use the distance formula
The formula for the distance \(d\) between the two lines represented by the equation is given by:
\[ d = \frac{2\sqrt{g^2 - ac}}{a + b} \]

### Step 3: Substitute the values into the formula
Now, substituting the values we identified:
- \(g = 13\)
- \(a = 8\)
- \(c = 15\)
- \(b = 2\)

We calculate:
\[ g^2 = 13^2 = 169 \]
\[ ac = 8 \times 15 = 120 \]
Thus,
\[ g^2 - ac = 169 - 120 = 49 \]

Now substituting into the distance formula:
\[ d = \frac{2\sqrt{49}}{8 + 2} = \frac{2 \times 7}{10} = \frac{14}{10} = \frac{7}{5} \]

### Step 4: Final expression for distance
The final expression for the distance between the two lines is:
\[ d = \frac{7}{2\sqrt{5}} \]

### Summary
Thus, the distance between the pair of straight lines represented by the given equation is:
\[ \frac{7}{2\sqrt{5}} \]