Question 2 - 2021 (16 Mar Shift 1)

Let the position vectors of two points $P$ and $Q$ be $3\hat{i} - \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} - 4\hat{k}$, respectively. Let $R$ and $S$ be two points such that the direction ratios of lines $PR$ and $QS$ are $(4, -1, 2)$ and $(-2, 1, -2)$, respectively. Let lines $PR$ and $QS$ intersect at $T$. If the vector $\overrightarrow{TA}$ is perpendicular to both $\overrightarrow{PR}$ and $\overrightarrow{QS}$ and the length of vector $\overrightarrow{TA}$ is $\sqrt{5}$ units, then the modulus of a position vector of $A$ is:

(1) $\sqrt{482}$

(2) $\sqrt{171}$

(3) $\sqrt{5}$

(4) $\sqrt{227}$