Browse Questions

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

$\begin {array} {1 1} (A)\;18x-12y-11=0& \quad (B)\;18x-12y+11=0 \\ (C)\;18x+12y+11=0 & \quad (D)\;-18x+12y+11=0 \end {array}$

Toolbox:
• The perpendicular distance between of a line from a point $(x_1, y_1)$ is $d= \bigg| \large\frac{Ax_1+By_1+c}{\sqrt{A^2+B^2}} \bigg|$
The equations of the given lines are :
$\qquad 9x+6y-7=0$--------(1) and
$\qquad 3x+2y+6=0$-------(2)
Let $p(a,b)$ be the point which is equidistant from lines (1) and (2)
The perpendicular distance of $p(a,b)$ from line (1) is given by
$d_1 = \large\frac{|9a+6b-7|}{\sqrt{9^2+6^2}}$
$= \large\frac{|9a+6b-7|}{\sqrt{117}}$
$= \large\frac{|9a+6b-7|}{3\sqrt{13}}$
Similarly the perpendicular distance of $p(a,b)$ from line (2) is
$d_2 = \large\frac{(3a+2b+6)}{\sqrt{(3)^2+(2)^2}}$
$=\large\frac{(3a+2b+6)}{\sqrt{13}}$
Since $p(a,b)$ is equidistant from line (1) and (2) we get, $d_1=d_2$
$\therefore \large\frac{|9a+6b-7|}{3\sqrt{13}}$$= \large\frac{|3a+2b+6|}{\sqrt{13}}$
$\Rightarrow |9a+6b-7| = 3|3a+2b+6|$
(i.e., ) $(9a+6b-7) = \pm 3(3a+2b+6)$ ( Considering the positive values)
(i.e.,) $(9a+6b-7) = 9a+6b+18$
This is not possible.
Hence $9a+6b-7 = -3(3a+2b+6)$
$\Rightarrow 9a+6b-7=-9a-6b-18$
$\Rightarrow 18a+12b+11=0$
Hence the required equation of the line is $18x+12y+11=0$