Question

Let \( PQR \) be a triangle such that \[ \vec{PQ}=-2\hat i-\hat j+2\hat k,\quad \vec{PR}=a\hat i+b\hat j-4\hat k,\ a,b\in\mathbb{Z}. \] Let \( S \) be the point on \( QR \) which is equidistant from the lines \( PQ \) and \( PR \). If \[ |\vec{PR}|=9 \quad \text{and} \quad \vec{PS}=\hat i-7\hat j+2\hat k, \] then the value of \( 3a-4b \) is:

Hint:

For equidistance from two lines through a point, always apply the angle-bisector condition using direction vectors.

Answer 1

Correct Answer: 29

Step 1: Understanding the Concept:
The point $S$ on $QR$ is equidistant from the lines $PQ$ and $PR$, which means $PS$ must be the interior angle bisector of $\angle QPR$.
Step 2: Key Formula or Approach:
Direction of angle bisector of $\vec{A}$ and $\vec{B}$ is $\frac{\vec{A}}{|\vec{A}|} + \frac{\vec{B}}{|\vec{B}|}$.
The point $S$ on the side $QR$ divides it in the ratio of the adjacent sides lengths: $QS/SR = |\vec{PQ}|/|\vec{PR}|$.
Step 3: Detailed Explanation:
$|\vec{PQ}| = \sqrt{(-2)^2 + (-1)^2 + 2^2} = 3$.
$|\vec{PR}| = \sqrt{a^2 + b^2 + 16} = 9 \implies a^2 + b^2 = 65$.
Since $S$ is on $QR$ and equidistant from the sides, $\vec{PS} = \frac{|\vec{PR}|\vec{PQ} + |\vec{PQ}|\vec{PR}}{|\vec{PR}| + |\vec{PQ}|}$.
$\vec{PS} = \frac{9(-2, -1, 2) + 3(a, b, -4)}{12} = \frac{3(-2, -1, 2) + (a, b, -4)}{4} = \left(\frac{a-6}{4}, \frac{b-3}{4}, \frac{2}{4}\right)$.
This leads to a mismatch in the $z$-coordinate ($1/2 \neq 2$). Thus, we reconsider $S$ using only the distance equality:
Distance from $S$ to line $PQ$: $d^2 = \frac{|\vec{PS} \times \vec{PQ}|^2}{|\vec{PQ}|^2} = \frac{|(1,-7,2) \times (-2,-1,2)|^2}{9} = \frac{|(-12,-6,-15)|^2}{9} = 45$.
Distance from $S$ to line $PR$: $d^2 = \frac{|(1,-7,2) \times (a,b,-4)|^2}{81} = 45$.
$|(28-2b)\hat{i} + (4+2a)\hat{j} + (b+7a)\hat{k}|^2 = 3645$.
Combined with $a^2 + b^2 = 65$, and $a, b \in \mathbb{Z}$, testing integer pairs $(7, -4)$ gives:
$| (36)\hat{i} + (18)\hat{j} + (45)\hat{k} |^2 = 36^2 + 18^2 + 45^2 = 1296 + 324 + 2025 = 3645$. (Matches).
Thus, $a = 7$ and $b = -4$.
Value $= 3a - 4b = 3(7) - 4(-4) = 21 + 16 = 37$.
Step 4: Final Answer:
The value of $3a - 4b$ is 37.