Question

The points, whose position vectors are $60i+3j,40i-8j$ and $ai-52j$ collinear, if

• A. $a=40$
• B. $a=-40$
• C. $a=20$
• D. $a=-20$

Answer 1

The correct option is B $a=-40$

Let $60i+3j,40i-8j,ai-52j$ be the position vectors of $A\left(\overline{a}\right)$,$B\left(\overline{b}\right)$ and $C\left(\overline{c}\right)$

For points A,B,C to be collinear

$\overline{AB}=\lambda \overline{BC}$,For some scalar $\lambda$

$\therefore$ $\overline{b}-\overline{a}=\lambda (\overline{c}-\overline{b})$

$(40i-8j)-(60i+3j)=\lambda \left[\right(ai-52j)-(40i-8j\left)\right]$
$-20=\lambda (a-40)$ and $-11=-44\lambda$
$\Rightarrow a-40=\frac{-20}{\lambda }$ and $\lambda =\frac{1}{4}$
$\Rightarrow a=40-\frac{20}{1}\times 4=-40$.