| Assertion (A): The point \[P(-4,6)\] divides the join of \[A(-6,10)\]and \[B(3,-8)\] in the ratio \[2:7\]. |
| Reason (R): If the point \[C(x,y)\] divides the join of \[A({{x}_{1}},{{y}_{1}})\] and \[B({{x}_{2}},{{y}_{2}})\]in the ratio \[m:n,\]then |
| \[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\] |
A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
B) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
C) Assertion (A) is true but reason (R) is false.
D) Assertion (A) is false but reason (R) is true.
Correct Answer: A
Solution :
| [a] Reason is clearly true. |
| Let \[P(-4,6)\]divides \[A(-6,10)\]and \[B(3,-8)\] in the ratio \[k:1\] |
| Then, \[\frac{k\times 3+1\times (-6)}{k+1}=-4\] |
| and \[\frac{k\times (-8)+1\times 10}{k+1}=6\] |
| \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3k-6=-4k-4\] |
| and \[-8k+10=6k+6\,\,\,\,\,\Rightarrow \,\,\,\,7k=2\] |
| and \[14k=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\,\,k=2/7\] |
| \[\therefore \] Required ratio is \[2/7:1\] i.e., \[2:7\] |
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