3D Geometry
Foot of perpendicular from a point to a line
nta_pyq_2023_jan
Grade 12
Question:
If the foot of the perpendicular drawn from $(1, 9, 7)$ to the line passing through the point $(3, 2, 1)$ and parallel to the planes $x + 2y + z = 0$ and $3y - z = 3$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to
Step-by-Step Solution
Key Concept: Find direction of line as cross product of plane normals, parametrize, use perpendicularity condition.
Direction of line $= (-5,1,3)$. Point on line: $(-5\lambda+3, \lambda+2, 3\lambda+1)$. Using $\overrightarrow{PM}\perp$ direction: $\lambda=1$, so $M=(-2,3,4)$. $\alpha+\beta+\gamma = -2+3+4 = 5$. Answer: (4)
Correct Answer: 5