3D Geometry
Shortest distance; parametric point on line
nta_pyq_2023_jan
Grade 12
Question:
Let the shortest distance between the lines $L: \frac{x-5}{-2} = \frac{y-\lambda}{0} = \frac{z+\lambda}{1}$, $\lambda \geq 0$ and $L_1: x+1 = y-1 = 4-z$ be $2\sqrt{6}$. If $(\alpha, \beta, \gamma)$ lies on L, then which of the following is NOT possible?
$\alpha + 2\gamma = 24$
$2\alpha + \gamma = 7$
$2\alpha - \gamma = 9$
$\alpha - 2\gamma = 19$
Step-by-Step Solution
Key Concept: Use SD formula with the two lines, solve for $\lambda$, then parametrize L and check which linear combination is impossible.
$\lambda=9$. Points on L: $\alpha=5-2t, \gamma=t-9$. $\alpha+2\gamma = 5-2t+2t-18 = -13 \neq 24$. So (1) is NOT possible. Answer: (1)
Correct Answer: 1