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Let $\vec{p}$ and $\vec{q}$ be the position vectors of the points $P$ and $Q$ respectively with respect to origin $\mathrm{O}$. The points $\mathrm{R}$ and $\mathrm{S}$ divide PQ internally and externally respectively in the ratio $2: 3 .$ If $\overrightarrow{\mathrm{OR}}$ and $\overrightarrow{\mathrm{OS}}$ are perpendicular, then which one of the following is correct?
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Verified Answer
The correct answer is:
$9 p^{2}=4 q^{2}$
R divides PQ internally in ratio $2: 3$
$\therefore \overrightarrow{\mathrm{OR}}=\frac{2 \overrightarrow{\mathrm{q}}+3 \overrightarrow{\mathrm{p}}}{5}$ ...(1)
S divides $\mathrm{PQ}$ externally in ratio $2: 3$
$\overrightarrow{\mathrm{OS}}=\frac{2 \overrightarrow{\mathrm{q}}-3 \overrightarrow{\mathrm{p}}}{2-3}=3 \overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}}$ ...(2)
Given, $\overrightarrow{\mathrm{OR}}$ and $\overrightarrow{\mathrm{OS}}$ are perpendicular.
$\therefore\left(\frac{3 \overrightarrow{\mathrm{p}}+2 \overrightarrow{\mathrm{q}}}{5}\right)(3 \overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}})=0$
$\Rightarrow 9 p^{2}-4 q^{2}=0 \Rightarrow 9 p^{2}=4 q^{2}$
$\therefore \overrightarrow{\mathrm{OR}}=\frac{2 \overrightarrow{\mathrm{q}}+3 \overrightarrow{\mathrm{p}}}{5}$ ...(1)
S divides $\mathrm{PQ}$ externally in ratio $2: 3$
$\overrightarrow{\mathrm{OS}}=\frac{2 \overrightarrow{\mathrm{q}}-3 \overrightarrow{\mathrm{p}}}{2-3}=3 \overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}}$ ...(2)
Given, $\overrightarrow{\mathrm{OR}}$ and $\overrightarrow{\mathrm{OS}}$ are perpendicular.
$\therefore\left(\frac{3 \overrightarrow{\mathrm{p}}+2 \overrightarrow{\mathrm{q}}}{5}\right)(3 \overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}})=0$
$\Rightarrow 9 p^{2}-4 q^{2}=0 \Rightarrow 9 p^{2}=4 q^{2}$
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