question_answer

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:

(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, it is showing 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, it is representing intersecting lines.