If the equation `ax^2+2hxy+by^2+2gx+2fy+c=0` represents a pair of parallel lines, prove that
`h=sqrt(ab) and gsqrt(b)=fsqrt(a)or (h=-sqrt(ab)and gsqrt(b)=-fsqrt(a))`.
The distance between them is `2sqrt((((g^2-ac))/(a(a+b))))`.

Answer

To prove that if the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) represents a pair of parallel lines, then \( h = \sqrt{ab} \) and \( g\sqrt{b} = f\sqrt{a} \) or \( h = -\sqrt{ab} \) and \( g\sqrt{b} = -f\sqrt{a} \), we can follow these steps:

### Step 1: Understand the Condition for Parallel Lines
For the equation to represent a pair of parallel lines, the discriminant of the quadratic in \( y \) (considering \( y \) as a variable) must be zero. This means that the lines represented by the equation must not intersect.

### Step 2: Write the Equation in Standard Form
The given equation can be expressed as:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
We can rearrange this to identify the coefficients of \( x \) and \( y \).

### Step 3: Find the Discriminant
For the quadratic equation in \( y \):
\[ by^2 + 2hxy + (ax^2 + 2gx + 2f + c) = 0 \]
The discriminant \( D \) is given by:
\[ D = (2hx)^2 - 4b(ax^2 + 2gx + 2f + c) \]
Setting \( D = 0 \) for parallel lines gives:
\[ 4h^2x^2 - 4b(ax^2 + 2gx + 2f + c) = 0 \]

### Step 4: Simplify the Equation
Factoring out \( 4 \):
\[ h^2x^2 - b(ax^2 + 2gx + 2f + c) = 0 \]
This implies:
\[ (h^2 - ab)x^2 - 2bgx - 2bc = 0 \]

### Step 5: Set Conditions for Parallel Lines
For this quadratic to represent parallel lines, the coefficients must satisfy:
1. \( h^2 = ab \)
2. \( g\sqrt{b} = f\sqrt{a} \) or \( g\sqrt{b} = -f\sqrt{a} \)

### Step 6: Conclusion
Thus, we have shown that if the equation represents a pair of parallel lines, then:
\[ h = \sqrt{ab} \quad \text{and} \quad g\sqrt{b} = f\sqrt{a} \quad \text{or} \quad h = -\sqrt{ab} \quad \text{and} \quad g\sqrt{b} = -f\sqrt{a} \]

### Step 7: Distance Between the Lines
The distance \( d \) between the two parallel lines can be calculated using the formula:
\[ d = \frac{2\sqrt{g^2 - ac}}{\sqrt{a(a+b)}} \]