Question

Let `overlinep` and `overlineq` be the position vectors of P and Q respectively, with respect to O and `|overlinep|` = P, `|overlineq|` = q`. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular, then ______

Options

• 9p² = 4q² 

• 4p² = 9q²

• 9p = 4q

• 4p = 9q

Answer 1

Let `overlinep` and `overlineq` be the position vectors of P and Q respectively, with respect to O and `|overlinep|` = P, `|overlineq|` = q`. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular, then 9p² = 4q².

Explanation:

R divides PQ internally in the ratio 2:3.

∴ `overliner = (2overlineq + 3overlinep)/8`

S divides PQ externally in the ratio 2:3.

∴ `overlines = (2overlineq - 3overlinep)/(2 - 3)`

⇒ `overlines = 3overlinep - 3overlineq`

OR and OS are perpendicular.

∴ `overliner . overlines = 0`

⇒ `2/5(-2)|overlineq|^2 + 3/5(3)|overlinep|^2 = 0`

⇒ `9/5p^2 = 4/5q^2`

⇒  9p² = 4q²