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:
$(2\hat{i}+aj+4\hat{k})\cdot [(x-2)\hat{i}+(y-a)\hat{j}+(z-4)\hat{k}]=0$
$\Rightarrow 2x+ay+4z=20+a^2$
$\Rightarrow A\equiv \left(\frac{20+a^2}{2},0,0\right)$
$B\equiv \left(0,\frac{20+a^2}{a},0\right)$
$C\equiv \left(0,0,\frac{20+a^2}{4}\right)$
$\Rightarrow$ 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 $2x+2y+4z=24$
Or $x+y+2z=12$
$\Rightarrow (3,0,4)$ Not lies on the Plane $x+y+2z=12$