Let vec{p} and vec{q} be the position vectors | vedclass

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(A) Let $\vec{r}$ and $\vec{s}$ be the position vectors of points $R$ and $S$ respectively.Using the section formula for internal division in ratio $2

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$9p^2 = 4q^2$

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(A) Let $\vec{r}$ and $\vec{s}$ be the position vectors of points $R$ and $S$ respectively.Using the section formula for internal division in ratio $2:3$:$\vec{r} = \frac{2\vec{q} + 3\vec{p}}{2+3} = \frac{3\vec{p} + 2\vec{q}}{5}$Using the section formula for external division in ratio $2:3$:$\vec{s} = \frac{2\vec{q} - 3\vec{p}}{2-3} = \frac{2\vec{q} - 3\vec{p}}{-1} = 3\vec{p} - 2\vec{q}$Since $\vec{OR}$ and $\vec{OS}$ are perpendicular,their dot product is zero:$\vec{r} \cdot \vec{s} = 0$$\left(\frac{3\vec{p} + 2\vec{q}}{5}\right) \cdot (3\vec{p} - 2\vec{q}) = 0$$(3\vec{p} + 2\vec{q}) \cdot (3\vec{p} - 2\vec{q}) = 0$$9|\vec{p}|^2 - 6(\vec{p} \cdot \vec{q}) + 6(\vec{q} \cdot \vec{p}) - 4|\vec{q}|^2 = 0$$9p^2 - 4q^2 = 0$$9p^2 = 4q^2$

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