​ On comparing the ratios a1/a2, b1/b2 and c1/c2 find out whether the lines representing ​

Question

On comparing the ratios \(\frac{a_1}{a_2}, \frac{b1}{b2} and \;\frac{c_1}{c_2}\) find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: 

(i) 5x – 4y + 8 = 0 

7x + 6y – 9 = 0 

(ii) 9x + 3y + 12 = 0

18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0

2x – y + 9 = 0

Answer 1

(i) 5x – 4y + 8 = 0 

a₁= 5, b₁ = -4, C₁= 8

7x + 6y – 9 = 0 

a₂ = 7, b₂= 6, c₂= -9

Here, \(\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\)

∴ Lines representing the pairs of linear equations intersect at a point. 

(ii) 9x + 3y + 12 = 0 

18x + 6y + 24 = 0 

a₁ =9, b₁ = 3, c₁ = 12 

a₂ = 18, b₂ = 6, c₂ = 24

\(\frac{c_1}{c_2} = \frac{12}{24} = \frac{1}{2}\)

Here, 

∴ Representation of lines graphically are coincident. 

(iii) 6x – 3y + 10 = 0 

2x – y + 9 = 0 

Here, 

a₁ = 6, b₁ = -3, c₁ = 10 

a₂ = 2, b₂ = -1, c₂ = 9

∴ Representation of lines graphically is parallel.