If O (origin) is a point inside the triangle PQR such that OP¯+k1OQ¯+k2OR¯=0¯ where k1and k2 are constants such that Area of triangle PQRArea of triangle OQR=4 then the positive value of k1+k2=______

Given that a→+k1b→+k2c→=0 Now, →Area of(ΔOQR)Area of(ΔPQR) =12(b→−a→)×(c→−a→)12|(b→×c→)|=4 ⇒a→×b→+b→×c→+c→×a→b→×c→=4⇒1+k1+k2=4Hence k1+k2=3

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If O (origin) is a point inside the triangle PQR such that $\bar{OP} + k_{1} \bar{OQ} + k_{2} \bar{OR} = \bar{0}$ where $k_{1} and k_{2}$ are constants such that $\frac{Area of triangle PQR}{Area of triangle OQR} = 4$ then the positive value of $k_{1} + k_{2} =$ ______

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answer is 3.

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Detailed Solution

Given that
$\vec{a} + k_{1} \vec{b} + k_{2} \vec{c} = 0 Now, \to_{Area of (ΔOQR)}^{Area of (ΔPQR)} = \frac{\frac{1}{2} |(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})|}{\frac{1}{2} | (\vec{b} \times \vec{c}) |} = 4$
Question Image
$\begin{array}{l} \Rightarrow \frac{|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|}{|\vec{b} \times \vec{c}|} = 4 \\ \Rightarrow 1 + k_{1} + k_{2} = 4 \\ Hence k_{1} + k_{2} = 3 \end{array}$