Question

Find the equation of the line mid-way between the parallel lines 9x + 6y − 7 = 0 and 3x + 2y + 6 = 0.

 

Answer 1

The given equations of the lines can be written as: 

\[3x + 2y - \frac{7}{3} = 0\]        ... (1)

3x + 2y + 6 = 0        ... (2)

Let the equation of the line midway between the parallel lines (1) and (2) be

\[3x + 2y + \lambda = 0\]   ... (3)

The distance between (1) and (3) and the distance between (2) and (3) are equal.

\[\therefore \left| \frac{- \frac{7}{3} - \lambda}{\sqrt{3^2 + 2^2}} \right| = \left| \frac{6 - \lambda}{\sqrt{3^2 + 2^2}} \right|\]

\[ \Rightarrow \left| - \left( \lambda + \frac{7}{3} \right) \right| = \left| 6 - \lambda \right|\]

\[ \Rightarrow 6 - \lambda = \lambda + \frac{7}{3}\]

\[ \Rightarrow \lambda = \frac{11}{6}\]

Equation of the required line:

\[3x + 2y + \frac{11}{6} = 0\]
\[ \Rightarrow 18x + 12y + 11 = 0\]