Question

Given the linear equation x - 2y - 6 = 0,write another linear equation in these two variables, such that the geometrical representation of the pair so formed is:
(i) coincident lines (ii) intersection lines

Answer 1

(i) Given, x - 2y - 6 = 0
For line to be coincident
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{C_1}{C_2}$
Thus one possible option will be
2x - 4y - 12 = 0
Here, $a_1=1,b_1=-2,c_1=-6$
$a_2=2,b_2=-4,c_2=-12$
$\frac{a_1}{a_2}=\frac{1}{2};\frac{b_1}{b_2}=\frac{-2}{-4}=\frac{1}{2};\frac{c_1}{c_2}=\frac{-6}{-12}=\frac{1}{2}$
$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
So, they are coincident lines.
(ii) Given, x - 2y - 6 = 0
For intersecting lines
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
Thus, one possible option will be,
$2x-7y-13=0$
Here, $a_1=1,b_1=-2,c_1=-6$
$a_2=2,b_2=-7,c_2=-13$
Here, $\frac{a_1}{a_2}=\frac{1}{2};\frac{b_1}{b_2}=\frac{-2}{-7}=\frac{2}{7}$
$\Rightarrow \frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
So, they are representing intersecting lines.