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, $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$