Question

The distance between the lines $3x+4y=9$ and $6x+8y=15$ is

• A. $\frac{3}{2}$
• B. $\frac{3}{10}$
• C. $6$
• D. None of these

Answer 1

Answer: (B)

If coefficient of $x$ and $y$ of both the lines are same, then the lines are parallel
Given, equation of lines are $3x+4y=9$ ... (i) and $6x+8y=15$
$\Rightarrow$ $3x+4y=\frac{15}{2}$ ... (ii)
$\therefore$ Both lines are parallel, therefore the distance between two lines $=\frac{|\frac{15}{2}-9|}{\sqrt{3^2+4^2}}$
$=\frac{|15-18|}{2\sqrt{25}}$
$=\frac{3}{2\cdot 5}=\frac{3}{10}$
Alternative The perpendicular distance from origin to the line
$L_1$ is
$d_1=\frac{9}{\sqrt{3^2+4^2}}=\frac{9}{5}$ and $d_2=\frac{\frac{15}{2}}{\sqrt{3^2+4^2}}$
$=\frac{15}{2\cdot 5}=\frac{15}{10}$
$\therefore$ Distance between $L_1$ and $L_2$ is $d=d_1-d_2$ $=\frac{9}{5}-\frac{15}{10}=\frac{18-15}{10}$
$=\frac{3}{10}$