$\begin {array} {1 1} (A)\;\large\frac{c_1-c_2}{\sqrt{m^2+1}} & \quad (B)\;\large\frac{|c_1-c_2|}{\sqrt{1+m^2}} \\ (C)\;\large\frac{c_2-c_1}{\sqrt {1+m^2}} & \quad (D)\;0 \end {array}$

- Distance between parallel lines is $ d = \bigg| \large\frac{c_1-c_2}{\sqrt{A^2+B^2}} \bigg|$

Step 1 :

The equation of the given lines are

$ y = mx+c_1$

$ \Rightarrow mx-y+c_1=0$ $ \therefore $ slope of this line is m and $y=mx+c_2$

$ \Rightarrow mx-y+c_2=0 \therefore $ slope of this line is m.

Since they have same slopes, the lines are parallel.

$ \therefore d = \bigg| \large\frac{c_1-c_2}{\sqrt{m^2+(-1)^2}} \bigg|$

$ = \bigg| \large\frac{ c_1-c_2}{\sqrt{1+m^2}} \bigg|$

Hence 'B' is the correct answer.

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