Question

The distance $d$ between two parallel lines $y=mx+c_1$ and $y=mx+c_2$ is given by $d=\frac{z}{\sqrt{1+m^2}}$, where $z$ stands for

• A. $c_1-c_2$
• B. $c_2-c_1$
• C. $\left|c_1+c_2\right|$
• D. $\left|c_1-c_2\right|$

Answer 1

Answer: (D)

Two parallel lines can be taken in the form
and $y=mx+c_1....$(i)
$y=mx+c_2....$(ii)
Line (i) will intersect $X$-axis at the point $A\left(-\frac{c_1}{m},0\right)$.
Distance between two lines is equal to the length of the perpendicular from point $A$ to line (ii). Therefore, distance between the lines (i) and (ii) is
$d=\frac{|(-m)\left(-\frac{c_1}{m}\right)+\left(-c_2\right)|}{\sqrt{1+m^2}}$
$d=\frac{|c_1-c_2|}{\sqrt{1+m^2}}$
Thus, the distanced between two parallel lines $y=mx+c_1$ and $y=mx+c_2$ is given by
$d=\frac{|c_1-c_2|}{\sqrt{1+m^2}}.$