Let A be a point on the line
Question:

Let $A$ be a point on the line

$\overrightarrow{\mathrm{r}}=(1-3 \mu) \hat{\mathrm{i}}+(\mu-1) \hat{\mathrm{j}}+(2+5 \mu) \hat{\mathrm{k}}$ and $\mathrm{B}(3,2,6)$

be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow{\mathrm{AB}}$ is parallel to the plane

$x-4 y+3 z=1$ is :

  1. $\frac{1}{2}$

  2. $-\frac{1}{4}$

  3. $\frac{1}{4}$

  4. $\frac{1}{8}$


Correct Option: , 3

Solution:

Let point $\mathrm{A}$ is

$(1-3 \mu) \hat{\mathrm{i}}+(\mu-1) \hat{\mathrm{j}}+(2+5 \mu) \hat{\mathrm{k}}$

and point $B$ is $(3,2,6)$

then $\overrightarrow{\mathrm{AB}}=(2+3 \mu) \hat{\mathrm{i}}+(3-\mu) \hat{\mathrm{j}}+(4-5 \mu) \hat{\mathrm{k}}$

which is parallel to the plane $x-4 y+3 z=1$

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

$8 \mu=2$

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

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