First slide

Planes in 3D

Question

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

Moderate

A

$V \leq \frac{9}{2}$
B

$V \geq \frac{9}{2}$
C

$V = \frac{9}{2}$
D

$V < \frac{9}{2}$

Solution

$Let the equation of the plane in intercept form be \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \dots \dots \dots \dots . (1)$
$Where a, b, c are positive intercepts on coordinate axes.$
$Eq. \underline{(1)}) is passing through (1, 1, 1)$
$\Rightarrow \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$
$We know that Volume of the tetrahedron O A B C = V = \frac{1}{6} (a b c)$
$Using the inequality Geometric mean (G.M.) \geq Harmonic mean (H.M.)$
$\begin{array}{l} \Rightarrow (a b c)^{\frac{1}{3}} \geq \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \\ \Rightarrow (a b c)^{\frac{1}{3}} \geq \frac{3}{1} \\ \Rightarrow a b c \geq 27 \\ \Rightarrow \frac{1}{6} (a b c) \geq \frac{1}{6} (27) \\ \Rightarrow V \geq \frac{9}{2} \end{array}$
Therefore, the correct answer is (2).

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