3D Geometry
Shortest Distance Between Two Lines
nta_pyq_2024_jan
Grade 12
Question:
The shortest distance between lines $L_1$ and $L_2$, where $L_1:\dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+4}{2}$ and $L_2$ is the line passing through the points $A(-4,4,3)$, $B(-1,6,3)$ and perpendicular to the line $\dfrac{x-3}{-2}=\dfrac{y}{3}=\dfrac{z-1}{1}$, is
$\dfrac{121}{\sqrt{221}}$
$\dfrac{24}{\sqrt{117}}$
$\dfrac{141}{\sqrt{221}}$
$\dfrac{42}{\sqrt{117}}$
Step-by-Step Solution
Key Concept: $L_2$ passes through $A(-4,4,3)$ and $B(-1,6,3)$: direction $(3,2,0)$. But must also be perpendicular to $(-2,3,1)$: $(3,2,0)\cdot(-2,3,1)=-6+6=0$ ✓. Apply SD formula between $L_1$ and $L_2$.
SD $=\dfrac{141}{\sqrt{221}}$.
Correct Answer: 3