| If the points \[M\,(1,-1),\]\[N\,(5,\,\,2)\] and \[C\,(k,\,\,5)\] are collinear, then k is equal to |
A) 6
B) \[-\,3\]
C) 9
D) 4
Correct Answer: C
Solution :
| Here, \[{{x}_{1}}=1,\]\[{{x}_{2}}=5,\]\[{{x}_{3}}=k,\]\[{{y}_{1}}=-1,\] |
| \[{{y}_{2}}=2\] and \[{{y}_{3}}=5\] |
| \[\therefore \] Points are collinear |
| Then, \[\Delta =0\] |
| \[\Rightarrow \] \[\{{{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})\}=0\] |
| \[\Rightarrow \] \[1\,(2-5)+5\,(5+1)+k\,(-1-2)=0\] |
| \[\Rightarrow \] \[-\,3+30-3k=0\] |
| \[\Rightarrow \] \[3k=27\]\[\Rightarrow \]\[k=9\] |
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