Question

Let $p$ and $q$ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $\left|p\right|=p,\left|q\right|=q$. The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2:3$ respectively. If $OR$ and $OS$ are perpendicular, then?

• A. $9p^2=4q^2$
• B. $4p^2=9p^2$
• C. $9p=4q$
• D. $4p=9q$

Answer 1

The correct option is A

$9p^2=4q^2$

Explanation for the correct option:

Equating the dot product of $R$ and $S$:

Given: points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2:3$ respectively.

$$\begin{gathered} R=\frac{(3p+2q)}{5} \\ S=(3p–2q) \end{gathered}$$

Also $OR$ and $OS$ are perpendicular

Therefore their dot product will be zero.

$$\begin{gathered} R.S.=0 \\ \Rightarrow \frac{1}{5}\left(3p+2q\right)·\left(3p-2q\right)=0 \\ \Rightarrow \left(3p+2q\right)·\left(3p-2q\right)=0 \\ \Rightarrow 9p^2-6pq+6pq-4p^2=0 \\ \Rightarrow 9p^2-4q^2=0 \\ \Rightarrow 9p^2=4q^2 \end{gathered}$$

Therefore, the correct answer is option (A).