Question

A line $L$ is passing through the point $P(0,1,-1)$ and perpendicular to both the lines $\frac{x-2}{2}=\frac{y-4}{1}=\frac{z+2}{4}$ and $\frac{x+2}{3}=\frac{y+4}{2}=\frac{z-2}{-2}.$ If the position vector of point $Q$ on line $L$ is $(a,b,c)$ such that $(PQ{)}^2=357,$ then possible value of $a+2b+3c$ is

• A. $26$
• B. $24$
• C. $-24$
• D. $7$

Answer 1

The correct option is B $24$

Equation of line $L$ is
$\overrightarrow{r}=(\hat{j}-\hat{k})+\lambda \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& 4\\ 3& 2& -2\end{array}\right|$
$\Rightarrow \overrightarrow{r}=(\hat{j}-\hat{k})+\lambda (-10\hat{i}+16\hat{j}+\hat{k})$
Point $Q$ on line $L$ is $(-10\lambda ,16\lambda +1,\lambda -1)$

Now, $(PQ{)}^2=357{\lambda }^2=357$
$\Rightarrow {\lambda }^2=1$
$\Rightarrow \lambda =\pm 1$
Possible position vectors of point $Q$ are $(-10,17,0)$ or $(10,-15,-2)$
$\Rightarrow a+2b+3c=24$ or $-26$