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
$3x+4y=9\dots (i)$
and $6x+8y=15$
$\Rightarrow 3x+4y=\frac{15}{2}\dots \left(ii\right)$
$\therefore$ Both lines are parallel, therefore the distance between two lines
$=\frac{|\frac{15}{2}-9|}{\sqrt{3^2+4^2}}$
$=\frac{|15-18|}{2\sqrt{25}}=\frac{3}{2.5}=\frac{3}{10}$
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 $d_2=\frac{2}{\sqrt{3^2+4^2}}$
$=\frac{15}{2\cdot 5}=\frac{15}{10}$
$\therefore$ Distance between $L_1$ and $L_2$ is
$d=d_1-d_2$
$=\frac{9}{5}-\frac{15}{10}=\frac{18+15}{10}$
$=\frac{3}{10}$