Vector Algebra
Tetrahedron volume; point not on a plane
nta_pyq_2023_jan
Grade 12
Question:
The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points A, B, C is $(2, a, 4)$, $a \in \mathbb{N}$. If the volume of the tetrahedron OABC is 144 unit$^3$, then which of the following points is NOT on P?
(2, 2, 4)
(0, 4, 4)
(3, 0, 4)
(0, 6, 3)
Step-by-Step Solution
Key Concept: Find plane equation from foot of perpendicular; plane: $2x+ay+4z=4+a^2+16$. Volume of tetrahedron $= abc/6$ where intercepts are $a_0,b_0,c_0$. Find $a$, test points.
$a=2$. Plane $x+y+2z=12$. Test (0,6,3): $0+6+6=12$ ✓. Test (2,2,4): $2+2+8=12$ ✓. Test (3,0,4): $3+0+8=11 \neq 12$ ✗. Answer: (3)
Correct Answer: (3, 0, 4)