| Find the value of k for which the points \[A\,\,(-1,3),\]\[B\,\,(2,k)\]and \[C\,\,(5,-1)\]are collinear |
A) 1
B) 3
C) 2
D) 4
Correct Answer: A
Solution :
| Here, \[{{x}_{1}}=-1,\]\[{{x}_{2}}=2,\]\[{{x}_{3}}=5,\]\[{{y}_{1}}=3,\]\[{{y}_{2}}=k\]and \[{{y}_{3}}=-1\] |
| Since, points are collinear, |
| Then, area \[(\Delta )=0\] |
| \[\Rightarrow \]\[{{x}_{1}}\,\,({{y}_{2}}-{{y}_{3}})+{{x}_{2}}\,\,({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})=0\] |
| \[\Rightarrow \]\[-1\,\,(k+1)+2\,\,(-1-3)+5\,\,(3-k)=0\] |
| \[\Rightarrow \]\[-\,\,k-1-8+15-5k=0\] |
| \[\Rightarrow \]\[6k=6\]\[\Rightarrow \]\[k=1\] |
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