| Equation of line \[{{L}_{1}}:{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0,\] |
| Equation of line \[{{L}_{2}}:{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0,\] |
| Equation of line \[{{L}_{3}}:({{a}_{1}}x+{{b}_{1}}y+{{c}_{1}})+({{a}_{2}}x+{{b}_{2}}y+{{c}_{2}})=0\] |
| if \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}},\] then line \[{{L}_{3}}\]is: |
A) parallel to line \[{{L}_{1}}\]
B) parallel to line \[{{L}_{2}}\]
C) is coincident with \[{{L}_{2}}\]or \[{{L}_{1}}\]
D) None of these
Correct Answer: C
Solution :
| [c] Lines \[{{L}_{1}}\] and \[{{L}_{2}}\] are coincident lines. |
| Line \[{{L}_{3}}:({{a}_{1}}+{{a}_{2}})x+({{b}_{1}}+{{b}_{2}})y+{{c}_{1}}+{{c}_{2}}=0\] |
| \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\] |
| \[\Rightarrow \,\,\,\,\,\,\,\frac{{{a}_{1}}}{{{a}_{2}}}+1=\frac{{{b}_{1}}}{{{b}_{2}}}+1=\frac{{{c}_{1}}}{{{c}_{2}}}+1\] |
| \[\Rightarrow \,\,\,\,\,\,\,\frac{{{a}_{1}}+{{a}_{2}}}{{{a}_{2}}}=\frac{{{b}_{1}}+{{b}_{2}}}{{{b}_{2}}}=\frac{{{c}_{1}}+{{c}_{2}}}{{{c}_{2}}}\] |
| \[\therefore \] \[{{L}_{3}}\] and \[{{L}_{2}}\] are coincident. |
| \[\therefore \] Lines \[{{L}_{1}},{{L}_{2}}\] and \[{{L}_{3}}\] are coincident. |
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