Question

Given the linear equation \(2x + 3y – 8 = 0\), write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines (ii) parallel lines (iii) coincident lines

Answer 1

 (i) Intersecting lines: 
For this condition, 
\(\dfrac{a_1}{a_2} ≠ \dfrac{b_1}{b_2}\)

The second line such that it is intersects the given line is 
\(2x +4y -6 =0\) as \(\dfrac{a_1}{a_2}= \dfrac{2}{2} =1\), \(\dfrac{b_1}{b_2} = \dfrac{3}{4}\) and \(\dfrac{a_1}{a_2} ≠ \dfrac{b_1}{b_2}\)

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(ii) Parallel lines: 
For this condition, 
\(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2}\)
Hence, the second line can be 
\(4x + 6y − 8 = 0\) as 

\(\dfrac{a_1}{a_2} =\dfrac{2}{4} = \dfrac{1}{2}, \dfrac{b_1}{b_2} = \dfrac{3}{6} = \dfrac{1}{2}, \dfrac{c_1}{c_2} = \dfrac{-8}{-8} = 1\)
And clearly, \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2}\)

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(iii) Coincident lines: 
For coincident lines, 
\(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\)

Hence, the second line can be 
\(6x + 9y − 24 = 0\) as 

\(\dfrac{a_1}{a_2} = \dfrac{2}{3} =\dfrac{1}{3}, \dfrac{b_1}{b_2} =\dfrac{3}{9} = \dfrac{1}{3} , \dfrac{c_1}{c_2}= \dfrac{-8}{-24} = \dfrac{1}{3}\)
And clearly, \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\)