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Question
The position vectors of points A, B and C are \[\lambda \hat{i} +\] 3 \[\hat{j}\],12\[\hat{i} + \mu\] \[\hat{j}\] and 11\[\hat{i} -\] 3 \[\hat{j}\] respectively. If C divides the line segment joining A and B in the ratio 3:1, find the values of \[\lambda\] and \[\mu\]
Solution
The position vectors of points A, B and C are \[\lambda \hat{i} +\] 3 \[\hat{j}\],12\[\hat{i} + \mu\] \[\hat{j}\] and 11\[\hat{i} -\] 3 \[\hat{j}\] respectively.
It is given that, C divides the line segment joining A and B in the ratio 3 : 1.
\[11 \hat{i} - 3 \hat{j} = \frac{3 \times \left( 12 \hat{i} + \mu \hat{j} \right) + 1 \times \left( \lambda \hat{i} + 3 \hat{j} \right)}{3 + 1}\]
\[ \Rightarrow 11 \hat{i} - 3 \hat{j} = \frac{\left( 36 + \lambda \right) \hat{i} + \left( 3\mu + 3 \right) \hat{j} }{4}\]
\[ \Rightarrow 44 \hat{i} - 12 \hat{j} = \left( 36 + \lambda \right) \hat{i} + \left( 3\mu + 3 \right) \hat{j}\]
Equating the corresponding components, we get \[36 + \lambda = 44\]
\[\Rightarrow \lambda = 44 - 36 = 8\]
and
\[\Rightarrow 3\mu + 3 = -12 \]
\[\Rightarrow 3\mu = - 12 - 3 = - 15\]
\[\Rightarrow \mu = - 5\]
Thus, the values of \[\lambda\] and \[\mu\] are 8 and −5, respectively.
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