Distance between the lines represented by `9x^2-6x y+y^2+18 x-6y+8=0` , is `1/(sqrt(10))` 2.`2/(sqrt(10))` 3.`4/(sqrt(10))` 4. `sqrt(10)` 5. `2/(sqrt(5))`

Options

A. `(1)/(sqrt(10))`
B. `(2)/(sqrt(10))`
C. `(4)/(sqrt(10))`
D. `sqrt(10)`

Answer

To find the distance between the lines represented by the equation \(9x^2 - 6xy + y^2 + 18x - 6y + 8 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared with the general form of a pair of straight lines:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the given equation, we identify:
- \(a = 9\)
- \(2h = -6 \Rightarrow h = -3\)
- \(b = 1\)
- \(2g = 18 \Rightarrow g = 9\)
- \(2f = -6 \Rightarrow f = -3\)
- \(c = 8\)

### 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, we substitute the values of \(a\), \(b\), \(g\), and \(c\) into the formula:
- \(g^2 = 9^2 = 81\)
- \(ac = 9 \times 8 = 72\)
- \(a + b = 9 + 1 = 10\)

Now substituting these values into the distance formula:
\[ d = \frac{2\sqrt{81 - 72}}{10} \]

### Step 4: Simplify the expression
Calculating inside the square root:
\[ 81 - 72 = 9 \]
Thus,
\[ d = \frac{2\sqrt{9}}{10} = \frac{2 \times 3}{10} = \frac{6}{10} = \frac{3}{5} \]

### Step 5: Final calculation
Now, we need to express this in terms of the square root:
\[ d = \frac{2}{\sqrt{10}} \]

### Conclusion
Thus, the distance between the lines represented by the given equation is:
\[ \boxed{\frac{2}{\sqrt{10}}} \]