3D Geometry 3 Question 46
46. If $P$ is $(3,2,6)$ is a point in space and $Q$ be a point on the line $\overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}})$. Then, the value of $\mu$ for which the vector $\mathbf{P Q}$ is parallel to the plane $x-4 y+3 z=1$, is
(2009)
(a) $\frac{1}{4}$
(b) $-\frac{1}{4}$
(c) $\frac{1}{8}$
(d) $-\frac{1}{8}$
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Answer:
Correct Answer: 46. (c)
Solution:
- Given, $\mathbf{O Q}=(1-3 \mu) \hat{\mathbf{i}}+(\mu-1) \hat{\mathbf{j}}+(5 \mu+2) \hat{\mathbf{k}}$ and $\mathbf{O P}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{k}$
[where, $O$ is origin]
Now, $\quad \mathbf{P Q}=(1-3 \mu-3) \hat{i}+(\mu-1-2) \hat{j}+(5 \mu+2-6) \hat{k}$
$ =(-2-3 \mu) \hat{i}+(\mu-3) \hat{j}+(5 \mu-4) \hat{k} $
$\because \quad \mathbf{P Q}$ is parallel to the plane $x-4 y+3 z=1$.
$\therefore \quad-2-3 \mu-4 \mu+12+15 \mu-12=0$
$\Rightarrow \quad 8 \mu=2 \Rightarrow \mu=\frac{1}{4}$
