Question

Let the equation of a line is $\frac{x - 2}{1}=\frac{y - 3}{2}=\frac{z - 4}{3}$ . An insect starts flying from $P\left(1 , 3 , 2\right)$ in a straight line meeting the given line at a point $R\left(a , b , c\right)$ and then goes to the point $Q\left(6 , 7 , 5\right)$ in a straight line such that $PR$ is perpendicular to $RQ.$ Then, the least value of $7\left(a + b + c\right)$ is equal to

• A. $105$
• B. $45$
• C. $10$
• D. $7$

Answer 1

Answer: (B)

Any general point on the line is $\left(\lambda + 2 , 2 \lambda + 3 , \lambda + 4\right)$

$\overrightarrow{P R}= < \lambda +1,2\lambda ,3\lambda +2>$
$\overrightarrow{R Q}= < 4-\lambda ,4-2\lambda ,1-3\lambda >$
From the diagram, $\overrightarrow{P R}\cdot \overrightarrow{R Q}=0$
$\Rightarrow \left(\lambda + 1\right)\left(4 - \lambda \right)+2\lambda \left(4 - 2 \lambda \right)+\left(3 \lambda + 2\right)\left(1 - 3 \lambda \right)=0$
$\Rightarrow \lambda ^{2}-3\lambda -4+4\lambda ^{2}-8\lambda +9\lambda ^{2}+3\lambda -2=0$
$\Rightarrow 14\lambda ^{2}-8\lambda -6=0$
$\Rightarrow \lambda =1,-\frac{3}{7}$
Coordinates of $R$ are $\left(3 , 5 , 7\right)$ or $\left(\frac{11}{7} , \frac{15}{7} , \frac{19}{7}\right)$
Least value of $7\left(a + b + c\right)=11+15+19=45$