Let $\mathrm{P}(2,1,5)$ be a point in space and Q be a point on the line $\bar{r}=(\hat{i}-\hat{j}+2…

Let $\mathrm{P}(2,1,5)$ be a point in space and Q be a point on the line $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})$. Then the value of $\mu$ for which the vector $\overline{\mathrm{PQ}}$ is parallel to the plane $3 x-y+4 z=1$ is
  1. $\frac{-16}{13}$
  2. $\frac{16}{13}$
  3. $-\frac{13}{16}$
  4. $\frac{2}{5}$

Solution

Given: - $P(2,1,5)$ - $Q=(1,2,5)+\mu(-3,1,5)$ - Plane equation: $3 x-y+4 z=1$
Finding $\mu$ : $\overrightarrow{P Q}=\langle-1-3 \mu, 1+\mu, 5 \mu\rangle$
Dot product with the normal vector of the plane $\langle 3,-1,4\rangle$ must be zero for $\overrightarrow{P Q}$ to be parallel to the plane: $\begin{gathered} 3(-1-3 \mu)-1(1+\mu)+4(5 \mu)=0 \\ -3-9 \mu-1-\mu+20 \mu=0 \\ 10 \mu-4=0 \Rightarrow \mu=\frac{4}{10}=\frac{2}{5} \end{gathered}$

Asked in: MHT CET 2024 (15 May Shift 1)

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