Q. A plane passing through (1, 1, 1) cuts positive direction of co-ordinate axes at $A,\ B$ and $C$, then the volume of tetrahedron $OABC$ satisfies
A) $V \leq \frac{9}{2}$
B) $V \geq \frac{9}{2}$
C) $V = \frac{9}{2}$
D) None of these
Explanation
(b)
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 $OABC = V = \frac{1}{6}(abc)$
Now $(abc)^{1/3} \geq \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \geq 3$ (G.M.$\geq$H.M.)
$\Rightarrow abc \geq 27\ \Rightarrow V \geq \frac{9}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.