The pair of equations \(x+2y-4=0\) and \(2x+4y-12=0\) represents two lines, which are?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Parallel
For two lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), the ratios of the coefficients decide how the lines lie. If \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\), the lines have the same slope but are different lines, so they are parallel and never meet.
\(\displaystyle\frac{a_1}{a_2}=\frac{1}{2}\) and \(\displaystyle\frac{b_1}{b_2}=\frac{2}{4}=\frac{1}{2}\), equal
\(\displaystyle\frac{c_1}{c_2}=\frac{-4}{-12}=\frac{1}{3}\), different from \(\frac{1}{2}\), so the lines are parallel
Why the other options are incorrect
Option (1) Intersecting at exactly one point — Incorrect. Two lines meet in one point only when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\), that is, when their slopes differ.
Option (2) Intersecting exactly two points — Incorrect. Two different straight lines can never meet in two points; they meet once or not at all.
Option (3) Coincident — Incorrect. Coincident lines need all three ratios equal, so that both equations give the same line; here \(\frac{c_1}{c_2}\) differs.