Will the Railway Tracks Cross?

 

❓ Concept Question

How can we determine whether two lines will intersect or remain parallel using the ratios of their coefficients?

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🖼️ Concept Image

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✍️ Short Concept

This concept is based on:

👉 Pair of Linear Equations in Two Variables

👉 Parallel and Intersecting Lines

👉 Comparison of Coefficients

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🔷 Step 1 — Write the Equations

The rails are represented by:

$$x+2y-4=0$$
$$2x+4y-12=0$$

Comparing with the standard form:

$$a_1x+b_1y+c_1=0$$
$$a_2x+b_2y+c_2=0$$

we get:

Coefficient	Equation 1	Equation 2
$a$	1	2
$b$	2	4
$c$	-4	-12

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🔷 Step 2 — Compare the Ratios

Calculate:

$$\frac{a_1}{a_2}=\frac{1}{2}$$
$$\frac{b_1}{b_2}=\frac{2}{4}=\frac{1}{2}$$
$$\frac{c_1}{c_2}=\frac{-4}{-12}=\frac{1}{3}$$

Thus,

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$$

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🔷 Step 3 — Identify the Nature of Lines

For two linear equations:

Case 1

$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$$

➡ Lines intersect at one point.

Case 2

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$$

➡ Lines are parallel.

Case 3

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$

➡ Lines are coincident.

Since:

$$\frac{1}{2}=\frac{1}{2}\ne\frac{1}{3}$$

the given lines are parallel.

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🔷 Step 4 — Conclusion About the Rails

Parallel lines never intersect.

Therefore, the two rails:

❌ Do not meet.

❌ Do not cross each other.

✔ Remain at a constant distance apart.

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🚨 Common Mistakes

❌ Comparing only $a_1/a_2$ and forgetting $c_1/c_2$.

❌ Using wrong signs while calculating ratios.

❌ Thinking equal slopes always mean coincident lines.

Remember:

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}$$

can mean either:

• Parallel lines, or
• Coincident lines

depending on $c_1/c_2$.

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✅ Final Takeaway

For the equations:

$$x+2y-4=0$$

and

$$2x+4y-12=0$$

we have:

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$$

Hence:

✔ The lines are parallel.

✔ The system is inconsistent.

✔ There is no solution.

✔ The two rails will not cross each other.

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⭐ Class 10 Insight

The ratio test is the fastest way to determine the nature of two straight lines:

Condition	Result
$\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$	Intersecting Lines
$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$	Parallel Lines
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$	Coincident Lines

This table is very important for CBSE Class 10 board exams.

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📚 Related Topics

• Find Zeroes and Verify Relationship Between Coefficients
• Find Quadratic Polynomial from Sum and Product of Zeroes
• Quadratic Polynomial with Irrational Zeroes Explained