Question

If points A (60$\hat{i}$ + 3$\hat{j}$), B (40$\hat{i}$ − 8$\hat{j}$) and C (a$\hat{i}$ − 52$\hat{j}$) are collinear, then a is equal to
(a) 40
(b) −40
(c) 20
(d) −20

Answer 1

(b) −40
Given: Three points $A\left(60\hat{i}+3\hat{j}\right),B\left(40\hat{i}-8\hat{j}\right)$ and $C\left(a\hat{i}-52\hat{j}\right)$ are collinear. Then, $\overrightarrow{AB}=\lambda \overrightarrow{BC}.$
We have,
$\overrightarrow{AB}=\left(40\hat{i}-8\hat{j}\right)-\left(60\hat{i}+3\hat{j}\right)=-20\hat{i}-11\hat{j}$
$\overrightarrow{BC}=\left(a\hat{i}-52\hat{j}\right)-\left(40\hat{i}-8\hat{j}\right)=\left(a-40\right)\hat{i}-44\hat{j}$
Therefore,
$$\begin{gathered} \overrightarrow{AB}=\lambda \overrightarrow{BC} \\ \Rightarrow -20\hat{i}-11\hat{j}=\lambda \left(a-40\right)\hat{i}-\lambda 44\hat{j} \\ \Rightarrow \lambda \left(a-40\right)=-20,-44\lambda =-11\Rightarrow \lambda =\frac{1}{4} \\ \Rightarrow a-40=-80 \\ \Rightarrow a=-40 \end{gathered}$$