Q.Find equation of the line which is equidistant from parallel lines $9x + 6y - 7 = 0$ and $3x + 2y + 6 = 0$.

Concept: Distance From Point To Line — the required line is the parallel line exactly halfway between the two given lines.

Step 1: Rewrite both lines with the same coefficients for $x$ and $y$.

First line: $9x + 6y - 7 = 0$

Second line: $3x + 2y + 6 = 0$ → multiply by 3: $9x + 6y + 18 = 0$

Step 2: For parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$, the equidistant line has constant term $\frac{C_1 + C_2}{2}$.

Here $C_1 = -7$, $C_2 = 18$, so …

Concept and Intuition

When two lines are parallel, the set of all points that are equidistant from both lines forms another line — the mid-parallel or the line of equidistance. This line is parallel to the given lines and lies exactly halfway between them.

Think of it like a road with two parallel curbs. The centre line of the road is equidistant from both curbs. That centre line is what we are finding.

The distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by:

For a line to be equidistant from two given parallel lines, every point on it must satisfy that its perpendicular distances to the two lines are equal. Since the required line is parallel to both, it has the same $A$ and $B$ coefficients, only a different constant term.

Step-by-step solution

1. Make the equations comparable

The two given lines are:

They are parallel because the ratios of coefficients of $x$ and $y$ are equal: $\frac{9}{3} = \frac{6}{2} = 3$. But the constant terms are different.

To find the mid-parallel, we must write both lines with the same $A$ and $B$ coefficients. Multiply $L_2$ by 3:

Now we have:

2. The form of the required line

Any line parallel to these has the form:

where $k$ is a constant to be determined.

3. Condition for equidistance

For any point $(x, y)$ on $L$, the perpendicular distances to $L_1$ and $L_2'$ must be equal. Since $L$ is parallel, we can instead compare the distances from any convenient point on $L$ — or more simply, compare the constant terms directly.

The distance from a point on $L$ to $L_1$ is:

But on $L$, $9x + 6y = -k$. So:

Similarly, distance to $L_2'$:

Setting $d_1 = d_2$: …