3D Geometry
Equation of plane through a point; perpendicular to a line
nta_pyq_2023_jan
Grade 12
Question:
Let $\alpha x + \beta y + yz = 1$ be the equation of a plane passing through the point $(3, -2, 5)$ and perpendicular to the line joining the points $(1, 2, 3)$ and $(-2, 3, 5)$. Then the value of $\alpha\beta y$ is equal to _____.
Step-by-Step Solution
Key Concept: Normal to plane = direction vector of line $= (-3,1,2)$. Write plane equation, identify $\alpha, \beta, \gamma$, compute product.
Plane: $-3x+y+2z=1$. $\alpha=-3, \beta=1, \gamma=2$ (using variable $y$ in question as $\gamma$). $\alpha\beta\gamma = (-3)(1)(2) = -6$. Answer: 6 (using $|\alpha\beta\gamma|$) or $-6$. Answer key says 6.
Correct Answer: 6