If three points A , B , C are collinear, whose position vectors a… - Vidyadip

MCQ

If three points $A , B , C$ are collinear, whose position vectors are $\hat{i}-2 \hat{j}-8 \hat{k}, 5 \hat{i}-2 \hat{k}$ and $11 \hat{ i }+3 \hat{ j }+7 \hat{ k }$ respectively, then the ratio in which B divides AC is

A
$1: 2$
✓
$2: 3$
C
$2: 1$
D
$1: 1$

✓

Answer

Correct option: B.

$2: 3$

(B) Let the point B divide AC in the ratio $\lambda: 1$
$\therefore \quad 5 \hat{ i }-2 \hat{ k }=\frac{\lambda(11 \hat{ i }+3 \hat{ j }+7 \hat{ k })+\hat{ i }-2 \hat{ j }-8 \hat{ k }}{\lambda+1}$
$\Rightarrow \lambda(5 \hat{ i }-2 \hat{ k })+(5 \hat{ i }-2 \hat{ k })$
$=\lambda(11 \hat{i}+3 \hat{j}+7 \hat{k})+(\hat{i}-2 \hat{j}-8 \hat{k})$
$\Rightarrow-6 \lambda \hat{ i }-3 \lambda \hat{ j }-9 \lambda \hat{ k }=-4 \hat{ i }-2 \hat{ j }-6 \hat{ k }$
$\Rightarrow-6 \lambda=-4$
$\Rightarrow \lambda=\frac{2}{3}$ i.e. ratio $=2: 3$

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