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Let $A (0, 6, 6), B (6, 6, 0)$ and $C (6, 0, 6)$ are three points and point $D$ is moving on the line $x + z - 3 = 0 = y$ . If $G$ is centroid of , then minimum value of $G D$ is

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$\sqrt{\frac{37}{2}}$

$\sqrt{\frac{23}{2}}$

$\sqrt{\frac{47}{2}}$

$\sqrt{\frac{57}{2}}$

answer is C.

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Detailed Solution

$G = (4, 4, 4)$

Let
$D = [a, 0, 3 - a]$ $G D^{2} = {(a - 4)}^{2} + {(0 - 4)}^{2} + {(3 - a - 4)}^{2} = a^{2} + 16 - 8 a + 16 + a^{2} + 1 + 2 a$

$= 2 a^{2} - 6 a + 33 = 2 (a^{2} - 3 a + a)$
For minimum ${GD}^{2}, a = \frac{3}{2}$
${GD}_{min} = \sqrt{2 \cdot \frac{9}{4} - 6 \cdot \frac{3}{2} + 33} = \sqrt{\frac{57}{2}}$

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