3D Geometry
Line Intersecting Two Lines and Parallel to Third
nta_pyq_2024_apr
Grade 12
Question:
Let the line $L$ intersect the lines $x-2=-y=z-1$, $2(x+1)=2(y-1)=z+1$ and be parallel to the line $\dfrac{x-2}{3}=\dfrac{y+1}{1}=\dfrac{z-2}{2}$. Then which of the following points lies on $L$?
$\left(-\dfrac{1}{3},1,-1\right)$
$\left(-\dfrac{1}{3},-1,1\right)$
$\left(-\dfrac{1}{3},1,1\right)$
$\left(-\dfrac{1}{3},-1,-1\right)$
Step-by-Step Solution
Key Concept: $L_1:\frac{x-2}{1}=\frac{y}{-1}=\frac{z-1}{1}=\lambda$, $L_2:\frac{x+1}{1/2}=\frac{y-1}{1/2}=\frac{z+1}{1}=\mu$. $M=(2+\lambda,-\lambda,1+\lambda)$ on $L_1$, $N=(-1+\mu/2,1+\mu/2,-1+\mu)$ on $L_2$. $L$ parallel to $(3,1,2)$.
Line $L$ through $(2/3,4/3,-1/3)$ parallel to $(3,1,2)$. Point $(-1/3,1,-1)$ lies on it.
Correct Answer: 1