Question

Let $\overline{p}$ and $\overline{q}$ be two distinct points. Let $R$ and $S$ be the points dividing $PQ$ internally and externally in the ratio $2:3$. If $\overline{OR}\perp \overline{OS},$ then

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

Answer 1

The correct option is A $9p^2=4q^2$

Position vector of $R=\frac{2\overline{q+3\overline{p}}}{5}$
position vector of $S=\frac{2\overline{q-3\overline{p}}}{(-1)}$
$=3\overrightarrow{p}-2\overline{q}$
Since $\overline{OR}\perp \overline{OS}$,

$\therefore \overline{OR}\cdot \overline{OS}=0$
$\therefore 9p^2=4q^2$