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)

Equation of Plane:
$ (2 \hat{ i }+ aj +4 \hat{ k }) \cdot[( x -2) \hat{ i }+( y - a ) \hat{ j }+( z -4) \hat{ k }]=0$
$ \Rightarrow 2 x + ay +4 z =20+ a ^2$
$ \Rightarrow A \equiv\left(\frac{20+ a ^2}{2}, 0,0\right) $
$ B \equiv\left(0, \frac{20+ a ^2}{ a }, 0\right) $
$ C \equiv\left(0,0, \frac{20+ a ^2}{4}\right)$
$\Rightarrow$ Volume of tetrahedron
$=\frac{1}{6}[\vec{a} \vec{b} \vec{c}]$
$ =\frac{1}{6} \vec{a} \cdot(\vec{b} \times \vec{c}) $
$ \Rightarrow \frac{1}{6}\left(\frac{20+ a ^2}{2}\right) \cdot\left(\frac{20+ a ^2}{ a }\right) \cdot\left(\frac{20+ a ^2}{4}\right)=144 $
$\Rightarrow\left(20+ a ^2\right)^3=144 \times 48 \times a$
$\Rightarrow a =2$
$\Rightarrow$ Equation of plane is $2 x +2 y +4 z =24$
Or $x+y+2 z=12$
$\Rightarrow(3,0,4)$ Not lies on the Plane $x+y+2 z=12$