The distance between the pair of parallel line, `x^2+2xy+y^2-8ax-8ay-9a^2=0` is

Options

A. `2 sqrt5 a `
B. `10 sqrta `
C. `10a`
D. `5sqrt2 a`

Answer

To find the distance between the pair of parallel lines represented by the equation \(x^2 + 2xy + y^2 - 8ax - 8ay - 9a^2 = 0\), we can follow these steps:

### Step 1: Rewrite the equation
The given equation can be rearranged as follows:
\[ x^2 + 2xy + y^2 - 8ax - 8ay - 9a^2 = 0 \]

### Step 2: Identify the quadratic form
This equation is a quadratic in \(x\) and can be expressed as:
\[ x^2 + (2y - 8a)x + (y^2 - 8ay - 9a^2) = 0 \]

### Step 3: Apply the quadratic formula
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 2y - 8a\), and \(c = y^2 - 8ay - 9a^2\):
\[ x = \frac{-(2y - 8a) \pm \sqrt{(2y - 8a)^2 - 4(1)(y^2 - 8ay - 9a^2)}}{2(1)} \]

### Step 4: Simplify the discriminant
Calculating the discriminant:
\[ (2y - 8a)^2 - 4(y^2 - 8ay - 9a^2) \]
Expanding this gives:
\[ 4y^2 - 32ay + 64a^2 - 4y^2 + 32ay + 36a^2 = 100a^2 \]

### Step 5: Substitute back into the quadratic formula
Now substituting back into the quadratic formula:
\[ x = \frac{8a - 2y \pm 10a}{2} \]
This results in two lines:
1. \(x = -y + 9a\)
2. \(x = -y - a\)

### Step 6: Confirm the lines are parallel
Both lines can be rewritten in slope-intercept form:
1. \(y = -x + 9a\)
2. \(y = -x - a\)

Since both lines have the same slope (\(-1\)), they are parallel.

### Step 7: Calculate the distance between the lines
The formula for the distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]
For our lines:
- The first line can be written as \(x + y - 9a = 0\) (thus \(C_1 = -9a\))
- The second line can be written as \(x + y + a = 0\) (thus \(C_2 = a\))

Substituting into the distance formula:
\[ d = \frac{|-9a - a|}{\sqrt{1^2 + 1^2}} = \frac{|-10a|}{\sqrt{2}} = \frac{10|a|}{\sqrt{2}} = 5\sqrt{2}|a| \]

### Final Answer
The distance between the pair of parallel lines is:
\[ 5\sqrt{2}|a| \]