Two rails are represented by the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0. Will the rails cross each other?
Step-by-Step Solution
Key Concept: For two linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, the lines intersect if $\frac{a_1}{a_2}<br>eq\frac{b_1}{b_2}$. If $\frac{a_1}{a_2}=\frac{b_1}{b_2}<br>eq\frac{c_1}{c_2}$ the lines are parallel (distinct) and do not meet. If all three ratios are equal, the lines coincide.
1. Write the given equations in the standard form $a x + b y + c = 0$:
\[\begin{aligned}
&\text{(i)}\; x + 2y - 4 = 0 \quad\Rightarrow\; a_1=1,\; b_1=2,\; c_1=-4,\\
&\text{(ii)}\; 2x + 4y - 12 = 0 \quad\Rightarrow\; a_2=2,\; b_2=4,\; c_2=-12.\end{aligned}\]
2. Compute the ratios of the coefficients:
\[\frac{a_1}{a_2}=\frac{1}{2},\qquad \frac{b_1}{b_2}=\frac{2}{4}=\frac{1}{2},\qquad \frac{c_1}{c_2}=\frac{-4}{-12}=\frac{1}{3}.\]
3. Compare the ratios:
- Since \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\) the two lines have the same slope; they are either coincident or parallel.
- However \(\frac{c_1}{c_2}=\frac{1}{3}
eq\frac{1}{2}=\frac{a_1}{a_2}\). Hence the third ratio is different.
4. Conclusion based on the condition:
- When \(\frac{a_1}{a_2}=\frac{b_1}{b_2}
eq\frac{c_1}{c_2}\), the lines are parallel distinct.
- Therefore the two rails do not intersect each other.
Answer: The rails do not cross; they are parallel distinct lines.
Correct Answer: No, the rails do not cross each other (the lines are parallel).