3D Geometry
Shortest Distance Between Skew Lines — Sum of α
nta_pyq_2026_jan
Grade None
Question:
The sum of all values of $\alpha$, for which the shortest distance between the lines $\dfrac{x+1}{\alpha}=\dfrac{y-2}{-1}=\dfrac{z-4}{-\alpha}$ and $\dfrac{z}{\alpha}=\dfrac{y-1}{2}=\dfrac{z-1}{2\alpha}$ is $\sqrt{2}$, is
Step-by-Step Solution
Key Concept: $\vec{b_1}\times\vec{b_2}=(0,-3\alpha^2,3\alpha)$. $|\vec{b_1}\times\vec{b_2}|=3|\alpha|\sqrt{\alpha^2+1}$. $(\vec{a_2}-\vec{a_1})=(1,-1,-3)$. $(\vec{a_2}-\vec{a_1})\cdot(\vec{b_1}\times\vec{b_2})=3\alpha^2-9\alpha$.
Sum of $\alpha=-6$.
Correct Answer: 4