Question

The points with position vectors $60\hat{i}+3\hat{j}$, $40\hat{i}-8\hat{j}$ and $a\hat{i}-52\hat{j}$ are collinear if

• A. $a=-40$
• B. $a=40$
• C. $a=20$
• D. None of these

Answer 1

Answer: (A)

Let $\overrightarrow{OA}=60\hat{i}+3\hat{j}$,
$\overrightarrow{OB}=40\hat{i}-8\hat{j}$
and $\overrightarrow{OC}=a\hat{i}-52\hat{j}$
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=20\hat{i}-11\hat{j}$
$\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\left(a-60\right)\hat{i}-55\hat{j}$
Now, $\overrightarrow{AB}\propto \overrightarrow{AC}$ as $A$, $B$, $C$ are collinear.
i.e. $\left(-20\hat{i}-11\hat{j}\right)=\lambda \left\{\left(a-60\right)\hat{i}-55\hat{j}\right\}$
On comparing, we get
$-20=\left(a-60\right)\lambda$ and $-11=-55\lambda$
$\Rightarrow \lambda =1/5$
$\therefore -20=\left(a-60\right)\left(1/5\right)$
$\Rightarrow a-60=-100$
$\Rightarrow a=-100+60$
$\Rightarrow a=-40$