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.
You need to login to perform this action.
You will be redirected in
3 sec