If the points A, B, C with position vectors \(20\hat{\imath}+\lambda\hat{\jmath},\ 5\hat{\imath}-\hat{\jmath}\) and \(10\hat{\imath}-13\hat{\jmath}\) respectively are collinear, then the value of \(\lambda\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −37
The points are collinear when \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are parallel, that is, their components are proportional:
\(\overrightarrow{AB} = (5 - 20)\hat{\imath} + (-1 - \lambda)\hat{\jmath} = -15\hat{\imath} - (1 + \lambda)\hat{\jmath}\)
\(\overrightarrow{BC} = (10 - 5)\hat{\imath} + (-13 + 1)\hat{\jmath} = 5\hat{\imath} - 12\hat{\jmath}\)
\( \displaystyle \frac{-15}{5} = \frac{-(1 + \lambda)}{-12} \)
\(-36 = 1 + \lambda\)
\(\lambda = -37\)