Vector Algebra
Angle Bisector on Triangle Side — Integer Answer
nta_pyq_2026_jan
Grade 12
Question:
Let $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{PR}=a\hat{i}+b\hat{j}-4\hat{k}$, $a,b\in\mathbb{Z}$. Let $S$ be the point on QR which is equidistant from the lines PQ and PR. If $|\overrightarrow{PR}|=9$ and $\overrightarrow{PS}=\hat{i}-7\hat{j}+2\hat{k}$, then the value of $3a-4b$ is ___.
Step-by-Step Solution
Key Concept: $|\overrightarrow{PR}|=9\Rightarrow a^2+b^2+16=81\Rightarrow a^2+b^2=65$. S equidistant from PQ and PR through P means PS is the angle bisector, so $\cos\theta_{PQ,PS}=\cos\theta_{PS,PR}$.
$3a-4b=37$.
Correct Answer: 37