On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{8}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$, find out whether the lines representing the
Question.
On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$, find out whether the lines representing the
following pairs of linear equations intersect at a point, are parallel or coincident.
(i) $5 x-4 y+8=0 ; 7 x+6 y-9=0$
(ii) $9 x+3 y+12=0 ; 18 x+6 y+24=0$
(iii) $6 x-3 y+10=0 ; 2 x-y+9=0$
On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$, find out whether the lines representing the
following pairs of linear equations intersect at a point, are parallel or coincident.
(i) $5 x-4 y+8=0 ; 7 x+6 y-9=0$
(ii) $9 x+3 y+12=0 ; 18 x+6 y+24=0$
(iii) $6 x-3 y+10=0 ; 2 x-y+9=0$
Solution:
(i) $5 x-4 y+8=0$ ...(i)
$7 x+6 y-9=0$ ...(ii)
$\frac{a_{1}}{a_{2}}=\frac{5}{7}, \frac{h_{1}}{b_{2}}=\frac{-4}{6}=-\frac{2}{3} \quad \Rightarrow \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
Lines represented by (i) and (ii) intersect at a point
(ii) $9 x+3 y+12=0$ ...(i)
$18 x+6 y+24=0$ ...(ii)
$\frac{a_{1}}{a_{2}}=\frac{9}{18}, \frac{b_{1}}{b_{2}}=\frac{3}{6}, \frac{c_{1}}{c_{2}}=\frac{12}{24}$
$\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Lines represented by (i) and (ii) are coincident
(iii) $6 x-3 y+10=0$ ...(i)
$2 x-y+9=0$ ...(ii)
$\frac{a_{1}}{a_{2}}=\frac{6}{2}=\frac{3}{1}, \frac{b_{1}}{b_{2}}=\frac{-3}{-1}=\frac{3}{1}, \frac{c_{1}}{c_{2}}=\frac{10}{9}$
$\Rightarrow \frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}=\frac{\mathbf{b}_{1}}{\mathbf{b}_{2}} \neq \frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$
Lines represented by (i) and (ii) are parallel
(i) $5 x-4 y+8=0$ ...(i)
$7 x+6 y-9=0$ ...(ii)
$\frac{a_{1}}{a_{2}}=\frac{5}{7}, \frac{h_{1}}{b_{2}}=\frac{-4}{6}=-\frac{2}{3} \quad \Rightarrow \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
Lines represented by (i) and (ii) intersect at a point
(ii) $9 x+3 y+12=0$ ...(i)
$18 x+6 y+24=0$ ...(ii)
$\frac{a_{1}}{a_{2}}=\frac{9}{18}, \frac{b_{1}}{b_{2}}=\frac{3}{6}, \frac{c_{1}}{c_{2}}=\frac{12}{24}$
$\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Lines represented by (i) and (ii) are coincident
(iii) $6 x-3 y+10=0$ ...(i)
$2 x-y+9=0$ ...(ii)
$\frac{a_{1}}{a_{2}}=\frac{6}{2}=\frac{3}{1}, \frac{b_{1}}{b_{2}}=\frac{-3}{-1}=\frac{3}{1}, \frac{c_{1}}{c_{2}}=\frac{10}{9}$
$\Rightarrow \frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}=\frac{\mathbf{b}_{1}}{\mathbf{b}_{2}} \neq \frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$
Lines represented by (i) and (ii) are parallel