Let p and q be the position vectors of P and Q respectively, with respect to O and…
Let p and q be the position vectors of P and Q respectively, with respect to O and $|\overline{\mathrm{p}}|=\mathrm{p},|\overline{\mathrm{q}}|=\mathrm{q}$. The points R and S divide PQ internally and externally in the ratio $2: 3$ respectively. If $O R$ and OS are perpendiculars, then
$9 \mathrm{p}^2=4 \mathrm{q}^2$
$4 \mathrm{p}^2=9 \mathrm{q}^2$
$9 \mathrm{p}=4 \mathrm{q}$
$4 \mathrm{p}=9 \mathrm{q}$
Solution
Let $\overline{\mathrm{r}}$ and $\overline{\mathrm{s}}$ be the position vectors of points R and $S$ respectively.
$\therefore \quad \overline{\mathrm{r}}=\frac{3 \mathrm{p}+2 \mathrm{q}}{3+2}$ and $\overline{\mathrm{s}}=\frac{3 \mathrm{p}-2 \mathrm{q}}{3-2}$
As OS and OR are perpendicular, we get $\overline{\mathrm{r}} \cdot \overline{\mathrm{s}}=0$
$\begin{aligned}
& \therefore \quad\left(\frac{3 p+2 q}{5}\right)\left(\frac{3 p-2 q}{1}\right)=0 \\
& \therefore \quad 9 p^2-6 p q+6 p q-4 q^2=0 \\
& \therefore \quad 9 p^2=4 q^2
\end{aligned}$