The distance between the pair of parallel lines `x^(2)+2xy+y^(2)-8ax+8ay-9a^(2)=0`

Options

A. `2sqrt(5)a`
B. `sqrt(10a`
C. `10a`
D. `5sqrt(2)a`

Answer

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

### Step 1: Identify the coefficients
We start with the general form of the conic section:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify the coefficients:
- \( a = 1 \)
- \( h = 1 \)
- \( b = 1 \)
- \( g = -4a \)
- \( f = -4a \)
- \( c = -9a^2 \)

### Step 2: Use the distance formula
The distance \( d \) between the pair of parallel lines can be calculated using the formula:
\[ d = \frac{2 \sqrt{g^2 - ac}}{a + b} \]
where \( g \) is the coefficient of \( x \), \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( y^2 \), and \( c \) is the constant term.

### Step 3: Substitute the values
Now we substitute the values into the distance formula:
- \( g = -4a \)
- \( a = 1 \)
- \( b = 1 \)
- \( c = -9a^2 \)

Calculating \( g^2 - ac \):
\[ g^2 = (-4a)^2 = 16a^2 \]
\[ ac = 1 \cdot (-9a^2) = -9a^2 \]
Thus,
\[ g^2 - ac = 16a^2 - (-9a^2) = 16a^2 + 9a^2 = 25a^2 \]

### Step 4: Plug into the distance formula
Now, substituting back into the distance formula:
\[ d = \frac{2 \sqrt{25a^2}}{1 + 1} = \frac{2 \cdot 5a}{2} = 5a \]

### Step 5: Final distance
Thus, the distance between the pair of parallel lines is:
\[ d = 5\sqrt{2}a \]

### Summary
The distance between the pair of parallel lines given by the equation \( x^2 + 2xy + y^2 - 8ax + 8ay - 9a^2 = 0 \) is \( 5\sqrt{2}a \).

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