The pair of linear equations \(mx+2y+3=0\) and \(3x+6y+2=0\) intersect each other, if
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(m\ne1\)
Two lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) intersect at one point when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\), that is, when they do not have the same slope.
\(\displaystyle \frac{a_1}{a_2}=\frac{m}{3}\) and \(\displaystyle \frac{b_1}{b_2}=\frac{2}{6}=\frac{1}{3}\)
The lines intersect when \(\displaystyle \frac{m}{3}\ne\frac{1}{3}\), that is, \(m\ne1\)
At \(m=1\) the two ratios are equal while \(\frac{c_1}{c_2}=\frac{3}{2}\) is not, so the lines are parallel and never meet
Tip: For a pair of lines, \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\) means intersecting; \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\) means parallel; all three ratios equal means the lines coincide.