Question

A plane passing through $(1,1,1)$ cuts positive direction of coordinate axes at $A,B$ and $C$, then the volume of tetrahedron $OABC$ satisfies-

• A. $V\le \frac{9}{2}$
• B. $V\ge \frac{9}{2}$
• C. $V=\frac{9}{2}$
• D. None of these.

Answer 1

The correct option is B $V\ge \frac{9}{2}$

Assume equation of plane is in intercept form, $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Given plane is passing through $(1,1,1)$
$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$
Volume of $OABC=v=\frac{1}{6}\left(abc\right)$
Now $(abc{)}^{1/3}\ge \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\ge 1$, since $G.M\ge A.M$
$\Rightarrow abc\ge 27\Rightarrow V\ge \frac{9}{2}$
The volume of tetrahedron $OABC$ $V=\frac{1}{6}abc$
$\Rightarrow V\ge \frac{9}{2}$