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QJEEM23Ma9016: JEE Main 2023 Physics - Class 11 JEEM23Ma9016 Solution: Let PQR be a triangle. The points A, B and C are on the sides…

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that

QAAR=RBBP=PCCQ=12{{QA} \over {AR}} = {{RB} \over {BP}} = {{PC} \over {CQ}} = {1 \over 2} . Then Area(ΔPQR)Area(ΔABC){{Area(\Delta PQR)} \over {Area(\Delta ABC)}} is equal to :

Multiple Choice Options

  • A 52\frac{5}{2}
  • B 4
  • C 2
  • D 3

Step-by-Step Verified Solution

Let the position vector of P,Q,RP, Q, R be 0→,a→,b→\overrightarrow{0}, \overrightarrow{a}, \overrightarrow{b}

⇒\Rightarrow Position vector of A=2a→+b→3,A=\frac{2 \overrightarrow{a}+\overrightarrow{b}}{3},

Position vector of B=2b→3B=\frac{2 \overrightarrow{b}}{3} and

Position vector of C=a→3C=\frac{\overrightarrow{a}}{3}
JEE Main 2023 (Online) 24th January Morning Shift Mathematics - Vector Algebra Question 75 English Explanation
amp;∴AB→=2b⃗3−(2a⃗+b⃗3)=b⃗3−2a⃗3=b⃗−2a⃗3amp;CA→=2a⃗+b⃗3−a⃗3=a⃗+b⃗3amp; Area of △PQR=12∣PQ→×PR→∣=12∣a⃗×b⃗∣amp; Area of △ABC=12∣CA→×AB→∣amp;=12∣(a⃗+b⃗3)×(b⃗−2a⃗3)∣=12∣a⃗×b⃗3∣amp;∴ Area (△PQR) Area (△ABC)=12∣a⃗×b⃗∣12∣a⃗×b⃗3∣=3 sq. units \begin{aligned}& \therefore \overrightarrow{A B}=\frac{2 \vec{b}}{3}-\left(\frac{2 \vec{a}+\vec{b}}{3}\right)=\frac{\vec{b}}{3}-\frac{2 \vec{a}}{3}=\frac{\vec{b}-2 \vec{a}}{3} \\\\& \overrightarrow{C A}=\frac{2 \vec{a}+\vec{b}}{3}-\frac{\vec{a}}{3}=\frac{\vec{a}+\vec{b}}{3} \\\\& \text { Area of } \triangle P Q R=\frac{1}{2}|\overrightarrow{P Q} \times \overrightarrow{P R}|=\frac{1}{2}|\vec{a} \times \vec{b}| \\\\& \text { Area of } \triangle A B C=\frac{1}{2}|\overrightarrow{C A} \times \overrightarrow{A B}| \\& =\frac{1}{2}\left|\left(\frac{\vec{a}+\vec{b}}{3}\right) \times\left(\frac{\vec{b}-2 \vec{a}}{3}\right)\right|=\frac{1}{2}\left|\frac{\vec{a} \times \vec{b}}{3}\right| \\\\& \therefore \frac{\text { Area }(\triangle P Q R)}{\text { Area }(\triangle A B C)}=\frac{\frac{1}{2}|\vec{a} \times \vec{b}|}{\frac{1}{2}\left|\frac{\vec{a} \times \vec{b}}{3}\right|}=3 \text { sq. units }\end{aligned}

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