3D Geometry
Point on Line at Given Distance
nta_pyq_2024_jan
Grade 12
Question:
A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a,b,c)$ where $a,b,c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a,b,c)$ and $Q(4,-12,3)$ is equal to
Step-by-Step Solution
Key Concept: Direction $AB=(12,4,6)\propto(6,2,3)$, magnitude $=7$. Unit vector $(6/7,2/7,3/7)$. At distance 21 from $A$: $P=A+21\cdot\hat{u}=(4+18,-6+6,-2+9)=(22,0,7)$. Since $a,b,c$ non-negative: $P=(22,0,7)$. Distance $PQ$.
$P=(22,0,7)$. $PQ=22$.
Correct Answer: 22