question_answer

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

Correct Answer: B

Solution :

Key Idea 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                 \[=\frac{\left| \frac{15}{2}-9 \right|}{\sqrt{{{3}^{2}}+{{4}^{2}}}}\]                 \[=\frac{|15-18|}{2\sqrt{25}}\]                 \[=\frac{3}{2\cdot 5}=\frac{3}{10}\] 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                 \[d={{d}_{1}}-{{d}_{2}}\]                     \[=\frac{9}{5}-\frac{15}{10}=\frac{18-15}{10}\]                     \[=\frac{3}{10}\]