Question

Assertion :Statement- 1: If the vectors $\overrightarrow{a}$ and $\overrightarrow{c}$ are non-collinear, then the lines $\overrightarrow{r}=6\overrightarrow{a}-\overrightarrow{c}+\lambda (2\overrightarrow{c}-\overrightarrow{a})$ and $\overrightarrow{r}=\overrightarrow{a}-\overrightarrow{c}+\mu (\overrightarrow{a}+3\overrightarrow{c})$ are coplanar. Reason: There exists $\lambda$ and $\mu$ such that the two values of $\overrightarrow{r}$ in statement 1 become same

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

It is given that :

$\overrightarrow{r_1}=(6-\lambda )\overrightarrow{a}+(2\lambda -1)\overrightarrow{c}$

$\overrightarrow{r_2}=(1+\mu )\overrightarrow{a}+(3\mu -1)\overrightarrow{c}$

Now if $\overrightarrow{r_1}=\overrightarrow{r_2}$

Then $6-\lambda =1+\mu$

$\mu +\lambda =5$ ...(i)

And

$2\lambda -1=3\mu -1$

$2\lambda =3\mu$ ...(ii)

Therefore,

$2\mu +2\lambda =10$ from (i)
$\therefore 5\mu =10$
$\mu =2$
$\lambda =3$ from (ii)

Hence for the above values of $\lambda$ and $\mu$, $r_1$ becomes equal to $r_2$, hence becoming co-planar.

Thus the reason is the correct explanation for the assertion.