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 equation of lines are

$3x+2y-\frac{7}{3}=0\dots \dots \left(i\right)$

$3x+2y+6=0\dots \dots \left(ii\right)$

Let equation of mid way be

$3x+2y+\lambda =0\dots \dots \left(iii\right)$

Then, distance between (i) and (iii) and (ii) and (iii) should be equal.

$\left|\frac{\lambda +\frac{7}{3}}{\sqrt{9+4}}\right|$

$\Rightarrow \lambda +\frac{7}{3}=-\pi +6$

$\Rightarrow \lambda =\frac{11}{6}$

$\therefore$ The required line is $3x+2y+\frac{11}{6}=0$ or $18x+12y+11=0$