If OABC is a tetrahedron (where O is origin and points $A (\bar{a}), B (\bar{b}), C (\bar{c})$ lies on X, Y, Z axes respectively). Let $A'$ is point on OA. $B'$ is point on OB & $C'$ is point on OC such that $A' B' C'$ plane is parallel to plane ABC and volume of tetrahedron $O A' B' C'$ is half the volume of tetrahedron OABC, then which of the following is/are CORRECT?

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Equation of plane through three points $A', B', C'$ is $\frac{x}{| \vec{a} |} + \frac{y}{| \vec{b} |} + \frac{z}{| \vec{c} |} = \frac{1}{2^{1 / 3}}$

Equation of plane through three points $A', B', C'$ is $\frac{x}{| \vec{a} |} + \frac{y}{| \vec{b} |} + \frac{z}{| \vec{c} |} = \frac{1}{2}$

$\frac{a r e a o f (Δ A B C)}{a r e a o f (Δ A' B' C')} = 2^{- 2 / 3}$

$\frac{a r e a o f (Δ A B C)}{a r e a o f (Δ A' B' C')} = 2^{- 1 / 3}$

answer is B.

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Detailed Solution

$\frac{[λ \vec{a} λ \vec{b} λ \vec{c}]}{6} = \frac{1}{6} \frac{[\vec{a} \vec{b} \vec{c}]}{2} λ = \frac{1}{2^{1 / 3}}$

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$\frac{x}{a λ} + \frac{y}{b λ} + \frac{z}{c λ} = 1 \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = \frac{1}{2^{1 / 3}} \frac{a r e a o f (Δ A B C)}{a r e a o f (Δ A' B' C')} = \frac{\frac{1}{2} \sqrt{{(a b)}^{2} + {(b c)}^{2} + {(c a)}^{2}}}{\frac{1}{2} \sqrt{λ^{2} [{(a b)}^{2} + {(b c)}^{2} + {(c a)}^{2}]}} = 2^{1 / 3}$