3D Geometry
Line through intersection of two lines
nta_pyq_2023_jan
Grade 12
Question:
Consider the lines $L_1: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-2}{2}$ and $L_2: \frac{x-2}{1} = \frac{y-2}{2} = \frac{z-3}{3}$. A line $L_3$ having direction ratios $1, -1, -2$, intersects $L_1$ and $L_2$ at the points P and Q respectively. Then the length of line segment PQ is
$2\sqrt{6}$
$3\sqrt{2}$
$4\sqrt{3}$
4
Step-by-Step Solution
Key Concept: Parametrize both lines, use direction of $L_3$ to set up equations for intersection points P and Q.
$P=(7,6,8)$ on $L_1$, $Q=(5,8,12)$ on $L_2$. $PQ = \sqrt{4+4+16} = \sqrt{24} = 2\sqrt{6}$. Answer: (1)
Correct Answer: $2\sqrt{6}$