Correct Answer - Option 2 :
\(\frac{17\sqrt 5}{15}\) unit
Concept:
The distance between the two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by:
\(d=\frac{\left | c_1-c_2 \right |}{\sqrt{a^2+b^2}}\)
Calculation:
The given equations are y = 2x + 4 and 6x = 3y + 5.
These can also be written as 2x - y + 4 = 0 and 2x - y - 5/3 = 0.
Now, since the lines are parallel.
\(\Rightarrow d=\frac{\left | c_1-c_2 \right |}{\sqrt{a^2+b^2}}\)
\(\Rightarrow d=\frac{\left | 4+\frac{5}{3} \right |}{\sqrt{2^2+3^2}}\)
\(\Rightarrow d=\frac{\left | \frac{17}{3} \right |}{\sqrt{5}}\)
\(\Rightarrow d=\frac{17\sqrt{5}}{15}\)
Hence, the distance between the parallel lines is \(\frac{17\sqrt 5}{15}\) unit.