# Parallel Pair of Straight Lines

## Condition Parallel Pair of Straight Lines

Consider a pair of straight lines given by the equation:

Writing it as a quadratic equation in \(x\):

Using the quadratic formula to solve for \(x\):

Simplifying under the square root:

There is a possibility that \(h^2 - ab = 0\) and \(hg - af = 0\) (while \(g^2 - ac\) may not be zero). In that case:

This gives us two values for \(x\):

These correspond to the equations:

Therefore, we get two parallel straight lines:

Thus, if \(h^2 - ab = 0\) and \(hg - af = 0\), the original quadratic equation represents two parallel straight lines.

So, the equation

represents a parallel pair of straight lines if

## Coincident Lines

If \(g^2 - ac = 0\) (which will also mean \(f^2 - bc = 0\)), then the lines are coincident.

When \(g^2 - ac = 0\), the equation for the lines simplifies as follows:

Given:

We previously found:

If \(g^2 - ac = 0\), the square root term vanishes:

This means both solutions for \(x\) are the same, indicating coincident lines. Specifically:

Similarly, considering the equation as a quadratic in \(y\):

We use the quadratic formula to solve for \(y\):

Simplifying under the square root:

For the expression inside the square root to be zero:

If \(h^2 - ab = 0\) and \(hf - ag = 0\), and also \(f^2 - ac = 0\), we get coincident lines.

So, the conditions for coincident lines are:

Under these conditions, the equation:

represents coincident lines. These conditions ensure that the quadratic equation simplifies into a single pair of overlapping lines, confirming that the lines are coincident.

The condition \( h^2 - ab = 0 \) indicates that the first three terms of the quadratic equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) form a perfect square. Specifically, this can be written as:

When \( h^2 - ab = 0 \), it means:

So the expression \( ax^2 + 2hxy + by^2 \) becomes:

Therefore, the quadratic equation can be rewritten as:

## Distance between a parallel pair of straight lines

Consider a parallel pair of straight lines represented by the quadratic equation:

Writing it as a quadratic equation in \(x\):

For the lines to be parallel, we set \(h^2 - ab = 0\) and \(hf - bg = 0\). Under these conditions, we get:

This simplifies to the equations of two parallel lines:

The distance \(d\) between these two parallel lines can be calculated using the formula for the distance between two parallel lines \(ax + by + c_1 = 0\) and \(ax + by + c_2 = 0\):

In this case, \(c_1 = g + \sqrt{g^2 - ac}\) and \(c_2 = g - \sqrt{g^2 - ac}\). Therefore, the distance \(d\) between the two lines is:

Substituting \(h^2 = ab\) in the denominator:

Thus, the distance between the two parallel lines is:

Alternatively,

Writing the given pair as a quadratic equation in \(y\):

For the lines to be parallel, we set \(h^2 - ab = 0\) and \(hf - bg = 0\). Under these conditions, we get:

This simplifies to the equations of two parallel lines:

The distance \(d\) between these two parallel lines can be calculated using the formula for the distance between two parallel lines \(ax + by + c_1 = 0\) and \(ax + by + c_2 = 0\):

Substituting \(h^2 = ab\) in the denominator:

Thus, the distance between the two parallel lines is:

The distance between the parallel straight lines represented by the quadratic equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) is:

or