Parallel, Intersecting and Coincident Lines Explained

 

❓ Question

On comparing the ratios:

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

find out whether the lines representing the following pairs of linear equations:

• intersect at a point
• are parallel
• or are coincident.

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(i)

$$5x-4y+8=0$$
$$7x+6y-9=0$$

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✍️ Short Concept

For equations:

$$a_1x+b_1y+c_1=0$$

and

$$a_2x+b_2y+c_2=0$$

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Case 1 — Coincident Lines

If:

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

then lines are coincident.

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Case 2 — Parallel Lines

If:

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

then lines are parallel.

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Case 3 — Intersecting Lines

If:

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

then lines intersect at one point.

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🔷 Step 1 — Compare Ratios

For:

$$5x-4y+8=0$$
$$a_1=5,\ b_1=-4,\ c_1=8$$

For:

$$7x+6y-9=0$$
$$a_2=7,\ b_2=6,\ c_2=-9$$

Now compare:

$$\frac{a_1}{a_2}=\frac57$$
$$\frac{b_1}{b_2}=\frac{-4}{6}$$

Since:

$$\frac57 \ne \frac{-4}{6}$$

the lines intersect at one point.

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✅ Final Answer (i)

$$\boxed{\text{Lines intersect at one point}}$$

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❓ Question (ii)

$$9x+3y+12=0$$
$$18x+6y+24=0$$

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🔷 Step 1 — Compare Ratios

For first equation:

$$a_1=9,\ b_1=3,\ c_1=12$$

For second equation:

$$a_2=18,\ b_2=6,\ c_2=24$$

Now:

$$\frac{a_1}{a_2}=\frac9{18}=\frac12$$
$$\frac{b_1}{b_2}=\frac36=\frac12$$
$$\frac{c_1}{c_2}=\frac{12}{24}=\frac12$$

All ratios are equal.

Hence lines are coincident.

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✅ Final Answer (ii)

$$\boxed{\text{Lines are coincident}}$$

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❓ Question (iii)

$$6x-3y+10=0$$
$$2x-y+9=0$$

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🔷 Step 1 — Compare Ratios

For first equation:

$$a_1=6,\ b_1=-3,\ c_1=10$$

For second equation:

$$a_2=2,\ b_2=-1,\ c_2=9$$

Now:

$$\frac{a_1}{a_2}=\frac62=3$$
$$\frac{b_1}{b_2}=\frac{-3}{-1}=3$$
$$\frac{c_1}{c_2}=\frac{10}{9}$$

Since:

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

the lines are parallel.

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✅ Final Answer (iii)

$$\boxed{\text{Lines are parallel}}$$

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⭐ Important Concept

For pair of linear equations:

• Intersecting lines → One solution
• Parallel lines → No solution
• Coincident lines → Infinite solutions

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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