Question

A plane passing through $(1,1,1)$ cuts positive direction of coordinate axes at $A,B$ anc $C,$ 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 $V\ge \frac{9}{2}$

Let the equation of the plane be $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$
$\Rightarrow$ Volume of tetrahedron $OABC$ $=V=\frac{1}{6}\left(abc\right)$

Now, using $G.M.\ge H.M.$
$(abc{)}^{1/3}\ge \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\ge 3$
$\Rightarrow abc\ge 27$
$\Rightarrow V\ge \frac{9}{2}$