# Choose the correct answer from the given four options. The ratio in which the line $3x + 4y + 2 = 0$ divides the distance between the lines $3x + 4y + 5 = 0$ and $3x + 4y – 5 = 0$ is

$\begin {array} {1 1} (A)\;1 : 2 & \quad (B)\;3 : 7 \\ (C)\;2 : 3 & \quad (D)\;2 : 5 \end {array}$

Toolbox:
• The distance between parallel line is $d = \bigg| \large\frac{c_1-c_2}{\sqrt{a^2+b^2}} \bigg|$
Step 1 :
The equation of the given lines are
$3x+4y+5=0$ ------(1) and
$3x+4y-5=0$-------(2)
Since both the lines have same slopes, the lines are parallel.
Equation of the line which divides the distances between these two lines is $3x+4y+2$--------(3)
This line also has the same slope.
Distance between lines (1) and line (3) is
$d_1 \bigg| \large\frac{5-2}{\sqrt{3^2+4^2}} \bigg|$$= \large\frac{3}{5}$
Distance between line (2) and line (3) is
$d_2 = \bigg| \large\frac{-5-2}{\sqrt{3^2+4^2}} \bigg|$
$= \large\frac{7}{5}$
Hence $d_1 : d_2 = 3 : 7$
$\therefore$ 'B' is the correct answer.