Question

Assertion :$L:\overrightarrow{r}=(\overrightarrow{i}+3\overrightarrow{j}-\overrightarrow{k})+t(\overrightarrow{j}+2\overrightarrow{k})$
$\pi :\overrightarrow{r}\cdot (\overrightarrow{i}+4\overrightarrow{j}+\overrightarrow{k})+6=0$

The line $L$ intersects the plane $\pi$ at the point $(1,0,-7)$

Reason: The angle between the line $L$ and the plane $\pi$ is $\left(\frac{1}{2}{\cos }^{-1}\left(\frac{1}{5}\right)\right)$

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$L:r=(i+3j-k)+t(j+2k)$
$\pi :r\cdot (i+4j+k)+6=0$
Let $\overrightarrow{OA}$ be a point that $L$ intersects at $\pi$
Therefore, $\overrightarrow{OA}=(i+3j-k)+t(j+2k)$
and $((i+3j-k)+t(j+2k))\cdot (i+4j+k)+6=0$
$\Rightarrow 1+12-1+t(1+2)=0$
$\Rightarrow t=-3$
Therefore, $\overrightarrow{OA}=(i+3j-k)-3(j+2k)=i-7k$
Angle between $L$ & $\pi$ is
$\sin \theta =\frac{(j+2k).(i+4j+k)}{\sqrt{(1^2+2^2)(1^2+4^2+1^2)}}=\frac{2}{\sqrt{10}}$
$\Rightarrow \cos 2\theta =1-2{\sin }^2\theta =1-\frac{8}{10}=\frac{1}{5}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{5}\right)$
Ans: B