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 $(2, a , 4), a \in N$ If the volume of the tetrahedron $OABC$ is 144 unit $^3$, then which of the following points is NOT on $P$ ?

• A. $(0,4,4)$
• B. $(3,0,4)$
• C. $(0,6,3)$
• D. $(2,2,4)$

Answer 1

Answer: (B)

$(3,0,4)$

Answer 2

Equation of Plane:
(2i^+aj+4k^)⋅[(x−2)i^+(y−a)j^​+(z−4)k^]=0
⇒2x+ay+4z=20+a2
⇒A≡(220+a2​,0,0)
B≡(0,a20+a2​,0)
C≡(0,0,420+a2​)
⇒ Volume of tetrahedron
=61​[abc]
=61​a⋅(b×c)
⇒61​(220+a2​)⋅(a20+a2​)⋅(420+a2​)=144
⇒(20+a2)3=144×48×a
⇒a=2
⇒ Equation of plane is 2x+2y+4z=24
Or x+y+2z=12
⇒(3,0,4) Not lies on the Plane x+y+2z=12
So , the correct option is (B) : $(3,0,4)$