3D Geometry
Shortest Distance Between Lines — Sum of λ Values
nta_pyq_2024_jan
Grade 12
Question:
If the shortest distance between the lines $\dfrac{x-\lambda}{-2}=\dfrac{y-2}{1}=\dfrac{z-1}{1}$ and $\dfrac{x-\sqrt{3}}{1}=\dfrac{y-1}{-2}=\dfrac{z-2}{1}$ is 1, then the sum of all possible values of $\lambda$ is:
0
$2\sqrt{3}$
$3\sqrt{3}$
$-2\sqrt{3}$
Step-by-Step Solution
Key Concept: Use SD formula: $\text{SD}=\frac{|(\vec{a_2}-\vec{a_1})\cdot(\vec{b_1}\times\vec{b_2})|}{|\vec{b_1}\times\vec{b_2}|}=1$. With $\vec{b_1}\times\vec{b_2}$ computed, find two values of $\lambda$.
$\lambda=0$ or $2\sqrt3$. Sum $=2\sqrt3$.
Correct Answer: 2