3D Geometry
Foot of perpendicular from a point to a line
nta_pyq_2023_jan
Grade 12
Question:
The foot of perpendicular of the point $(2, 0, 5)$ on the line $\frac{x+1}{2} = \frac{y-1}{5} = \frac{z+1}{-1}$ is $(\alpha, \beta, \gamma)$. Then which of the following is NOT correct?
\frac{\alpha\beta}{\gamma} = \frac{4}{15}
\frac{\alpha}{\beta} = -8
\frac{\beta}{\gamma} = -5
\frac{\gamma}{\alpha} = \frac{5}{8}
Step-by-Step Solution
Key Concept: Parametrize line, find foot using perpendicularity, then verify each option.
$\alpha=-4/3, \beta=1/6, \gamma=-5/6$. Check: $\alpha/\beta = (-4/3)/(1/6) = -8$ (correct). $\beta/\gamma = (1/6)/(-5/6) = -1/5 \neq -5$. So (3) is NOT correct. Answer: (3)
Correct Answer: 3