Question

Let $P(3,2,6)$ be point in space and $Q$ be appoint 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. $1/4$
• B. $-1/4$
• C. $1/8$
• D. $-1/8$

Answer 1

The correct option is A $1/4$

Consider the problem

Let

$\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu (-3\hat{i}+\hat{j}+5\hat{k})$

Then,

$\overrightarrow{r}=(1-3\mu ).\hat{i}+(-1+\mu ).\hat{j}+(2+5\mu ).\hat{k}$

Coordinates of $Q$ are

$(1-3\mu ),(-1+\mu ),(2+5\mu )$

Therefore direction ratios of $PQ$ are

$(-2-3\mu ,-3+\mu ,-4+5\mu )$

And since, $PQ$ is parallel to $x-4y+3z=1$ ---- $\left(i\right)$

Direction ratios of line which parallel to line $\left(i\right)$

given by:

$1.(-2-3\mu )-4.(-3+\mu ,)+3.(-4+5\mu )=0$

$-2-3\mu +12+4\mu -12+15\mu =0$

$8\mu -2=0$

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