Question

Let $P(3,2,6)$ be a point in space and $Q$ be a point on the line $\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu (-3\hat{i}+\hat{j}+5\hat{k})$. Then the value of $\mu$ for which the vector $\overrightarrow{PQ}$ is parallel to the plane $x-4y+3z=1$ is:

• A. $\frac{1}{4}$
• B. $-\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $-\frac{1}{8}$

Answer 1

The correct option is A $\frac{1}{4}$

Any point on the vector $\overrightarrow{r}$ can be taken as,
$Q\equiv \left\{\right(1-3\mu ),(\mu -1),(5\mu +2\left)\right\}$ gives

$\overrightarrow{PQ}=\{-3\mu -2,\mu -3,5\mu -4\}$
Now, the $\overrightarrow{PQ}$ must be perpendicular to the normal for the given plane.

$1(-3\mu -2)-4(\mu -3)+3(5\mu -4)=0\Rightarrow -3\mu -2-4\mu +12+15\mu -12=0\Rightarrow 8\mu =2$

$\Rightarrow \mu =\frac{1}{4}$