Question

If the points $(0,1.-2),(3,\lambda ,-1)$ and $(\mu ,-3,-4)$ are collinear, then the value of $\lambda$ and $\mu$ are given by

• A. $\lambda =3,\mu =-6$
• B. $\lambda =3,\mu =6$
• C. $\lambda =-3,\mu =6$
• D. $\lambda =-3,\mu =-6$

Answer 1

Answer: (A)

$\lambda =3,\mu =-6$

Consider the problem

Given points

$A(0,1.-2),B(3,\lambda ,-1),C(\mu ,-3,-4)$

$\overrightarrow{AB}=3\overrightarrow{i}+(\lambda -1)\overrightarrow{j}+\overrightarrow{k}$

$\overrightarrow{AC}=\mu \overrightarrow{i}-4\overrightarrow{j}-2\overrightarrow{k}$

Given points are collinear if

$\overrightarrow{AB}=k\overrightarrow{AC}$

where $k$ is any scalar

$3\overrightarrow{i}+(\lambda -1)\overrightarrow{j}+\overrightarrow{k}=k\left(\mu \overrightarrow{i}-4\overrightarrow{j}-2\overrightarrow{k}\right)$

then,

$\begin{array}{c}3=k\mu ----\left(1\right)\\ \lambda -1=-4k----\left(2\right)\\ 1=-2k----\left(3\right)\end{array}$

From $\left(3\right)$

$k=-\frac{1}{2}$

Putting $k=-\frac{1}{2}$ in $\left(1\right)$

we get,

$\mu =-6$

Putting $k=-\frac{1}{2}$ in $\left(2\right)$

We get,

$\lambda =3$

Therefore,
$\lambda =3,\mu =-6$