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, then the value of $a$ is

• A. $-40$
• B. $10$
• C. $20$
• D. $-30$

Answer 1

The correct option is A $-40$

Let $P,Q$ and $R$ be the collinear points with position vectors $60\hat{i}+3\hat{j},40\hat{i}-8\hat{j}$ and $a\hat{i}-52\hat{j}$ respectively.
Then $\overrightarrow{PQ}=\lambda \overrightarrow{QR},$ for some scalar $\lambda$
$\Rightarrow -20\hat{i}-11\hat{j}=\lambda \left(\right(a-40)\hat{i}-44\hat{j})$
$\Rightarrow \lambda (a-40)=-20$ and $-44\lambda =-11$
Hence we have $\lambda =\frac{1}{4}$ and $a=-40$