Question

The distance between the lines y = mx + c₁ and y = mx + c₂, is

(a) $\frac{c_1-c_2}{\sqrt{1+m^2}}$

(b) $\frac{\left|c_1-c_2\right|}{\sqrt{1+m^2}}$

(c) $\frac{c_2-c_1}{\sqrt{1+m^2}}$

(d) $\left|c_1-c_2\right|$

Answer 1

Consider a point P(x₁, y₁) an y = mx + c₁ and then distance between P(x₁, y₁) and y = mx + c₂ is same as distance between y = mx + c₁ and y = mx + c₂
i.e Distance between point P from the line,

$$\begin{gathered} d=\frac{\left|m{x}_1-y_1+c_2\right|}{\sqrt{m^2+1}} \\ \mathrm{Since}{y}_1=m{x}_1+c_1 \\ \mathrm{We}\mathrm{have}m{x}_1-y_1=-c_1 \\ \mathrm{i}.\mathrm{e}.d=\frac{\left|-c_1+c_2\right|}{\sqrt{m^2+1}} \\ \mathrm{i}.\mathrm{e}.d=\frac{\left|c_1-c_2\right|}{\sqrt{m^2+1}} \\ \therefore \mathrm{distance}\mathrm{between}y=mx+c_1\mathrm{and}y=mx+c_2\mathrm{is}\frac{\left|c_1-c_2\right|}{\sqrt{m^2+1}} \end{gathered}$$

Hence, the correct answer is option B.