Let p and q be the position vectors of P and Q respectively with respect to O and `|p| = p, |q| = q`. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If `vec( OR)` and `vec(OS)` are perpendicular, then (A) `9p^2=4q^2` (B) `4p^2=9q^2` (C) `9p=4q` (D) `4p=9q`

Options

A. `9p^(2)=4q^(2)`
B. `4p^(2)=9p^(2)`
C. `9p=4q`
D. `4p=9q`

Answer

A

Explanation

R divides PQ internally in ratio 2:3

`:.vec(OR)= (2vec(q)+3vec(p))/(5) ...(1)`
S divides PQ externally in ratio 2:3
`vec(OS)=(2vec(q)+3vec(p))/(2-3)=3vec(p)-2vec(q) ...(2)`
Given, `vec(OR) and vec(OS)` are perpendicular.
`:. ((3vec(p)+2vec(q))/(5))(3vec(p)-2vec(q))=0`
`rArr 9 p^(2)-4q^(2)=0rArr 9p^(2)=4q^(2)`