Question

The ratio in which the line $3x + 4y + 2 = 0$ divides the distance between $3x + 4y + 5 = 0$ and $3x + 4y - 5 = 0$ is
1. $7:3$
2. $3:7$
3. $2:3$
4. None of these

Answer 1

The distance between two parallel lines in the plane is the minimum distance between any two points lying on the lines. It equals the perpendicular distance from any point on one line to the other line. Find the respective distances between the lines using the formula $d = \left| {\dfrac{{{c_2} - {c_1}}}{{\sqrt {{a^2} + {b^2}} }}} \right|$ and then divide them accordingly to find the required ratio.

Complete step-by-step solution:
The lines can be extended till infinity. The slopes of two parallel lines are equal.
When the two lines are given by $ax + by + {c_1} = 0$ and $ax + by + {c_2} = 0$ then the distance between them can be expressed as $d = \left| {\dfrac{{{c_2} - {c_1}}}{{\sqrt {{a^2} + {b^2}} }}} \right|$
The lines $3x + 4y + 2 = 0$ and $3x + 4y + 5 = 0$ are on the same side of the origin. The distance between these lines is ${d_1} = \left| {\dfrac{{2 - 5}}{{\sqrt {{3^2} + {4^2}} }}} \right| = \dfrac{3}{5}$
The lines $3x + 4y + 2 = 0$ and $3x + 4y - 5 = 0$are on the opposite sides of the origin. The distance between these lines is ${d_2} = \left| {\dfrac{{2 + 5}}{{\sqrt {{3^2} + {4^2}} }}} \right| = \dfrac{7}{5}$
Thus , the line $3x + 4y + 2 = 0$ divides the distance between $3x + 4y + 5 = 0$ and $3x + 4y - 5 = 0$ in the ratio ${d_1}:{d_2} = 3:7$.
Hence we get the required ratio.
Therefore option (4) is the correct answer.

Note: Construct a graphical representation of the lines for better understanding. Use the formula for distance between lines correctly. Keep in mind that parallel lines are those lines that never meet each other. When the distance between a pair of lines is the same throughout, it can be called parallel lines. It is denoted by “||”. The main criteria for any two lines to be parallel is that they have to be drawn on the same plane. They are always equidistant from each other. Find the respective distances between the lines and then divide them accordingly to find the ratio. Do the calculations correctly so as to get the correct solution.