Three Dimensional Geometry Question 404
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
Options:
A) $ V\le \frac{9}{2} $
B) $ V\ge \frac{9}{2} $
C) $ V=\frac{9}{2} $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
[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}}\ge \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\ge 3(GM.\ge H.M.) $ or $ abc\ge 27\Rightarrow V\ge \frac{9}{2} $
BETA
