Question

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

• A. Statement-$1$ is true, statement-$2$ is true $1$ statement-$2$ is correct explanation for statement-$1$.
• B. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is not correct explanation for statement-$1$.
• C. Statement-$1$ is true, statement-$2$ is false.
• D. Statement-$1$ is false, statement-$2$ is true

Answer 1

The correct option is A Statement-$1$ is true, statement-$2$ is true $1$ statement-$2$ is correct explanation for statement-$1$.

Since a and c are not collinear, we can equate the coefficients to find $\mu$ and $\lambda$

Equating $\overrightarrow{r}$

$6\overrightarrow{a}-\overrightarrow{c}+\lambda (2\overrightarrow{c}-\overrightarrow{a})=\overrightarrow{a}-\overrightarrow{c}+\mu (\overrightarrow{a}+3\overrightarrow{c})$

$5\overrightarrow{a}+2\lambda \overrightarrow{c}-\lambda \overrightarrow{a}=\mu \overrightarrow{a}+3\mu \overrightarrow{c}$

Equating the coefficients of $\overrightarrow{a}$ and $\overrightarrow{c}$, we get

$5-\lambda =\mu$ ......... $\left(1\right)$

$2\lambda =3\mu$ ...... $\left(2\right)$

Solving equations $\left(1\right)$ and $\left(2\right)$, we get

$\lambda =3$ and $\mu =2$

Hence, statement 1 and 2 are true and statement 2 is correct explanation for statement 1.