Classify the following pairs of lines as coincident, parallel or inter

1.	Classify the following pairs of lines as coincident, parallel or intersecting: (i) 2x + y – 1 = 0 and 3x + 2y + 5 = 0 (ii) x – y = 0 and 3x – 3y + 5 = 0 (iii) 3x + 2y – 4 = 0 and 6x + 4y – 8 = 0
Answer» Let a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 be the two lines. (a) The lines intersect if \(\frac{a_1}{a_2}≠\frac{b_1}{b_2}\) is true. (b) The lines are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}\) is true. (c) The lines are coincident if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) is true. (i) Given: 2x + y − 1 = 0 and 3x + 2y + 5 = 0 Explanation: Here, \(\frac{2}{3}≠\frac{1}{2}\) Therefore, the lines 2x + y − 1 = 0 and 3x + 2y + 5 = 0 intersect. (ii) Given: x − y = 0 and 3x − 3y + 5 = 0 Explanation: Here, \(\frac{1}{3}=-\frac{1}{-3}≠\frac{0}{5}\) Therefore, the lines x − y = 0 and 3x − 3y + 5 = 0 are parallel. (iii) Given: 3x + 2y − 4 = 0 and 6x + 4y − 8 = 0 Explanation: Here, \(\frac{3}{6}=\frac{2}{4}=-\frac{4}{-8}\) Therefore, the lines 3x + 2y − 4 = 0 and 6x + 4y − 8 = 0 are coincident.

Classify the following pairs of lines as coincident, parallel or intersecting: (i) 2x + y – 1 = 0 and 3x + 2y + 5 = 0 (ii) x – y = 0 and 3x – 3y + 5 = 0 (iii) 3x + 2y – 4 = 0 and 6x + 4y – 8 = 0

Answer»

Let a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 be the two lines.

(a) The lines intersect if \(\frac{a_1}{a_2}≠\frac{b_1}{b_2}\) is true.

(b) The lines are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}\) is true.

(i) Given: 2x + y − 1 = 0 and 3x + 2y + 5 = 0

Explanation:

Here,

\(\frac{2}{3}≠\frac{1}{2}\)

Therefore, the lines 2x + y − 1 = 0 and 3x + 2y + 5 = 0 intersect.

(ii) Given: x − y = 0 and 3x − 3y + 5 = 0

Explanation:

Here, \(\frac{1}{3}=-\frac{1}{-3}≠\frac{0}{5}\)

Therefore, the lines x − y = 0 and 3x − 3y + 5 = 0 are parallel.

(iii) Given: 3x + 2y − 4 = 0 and 6x + 4y − 8 = 0

Explanation:

Here, \(\frac{3}{6}=\frac{2}{4}=-\frac{4}{-8}\)

Therefore, the lines 3x + 2y − 4 = 0 and 6x + 4y − 8 = 0 are coincident.