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Let O be origin and point $P_{1} (1, 1, 1)$ lie on $L_{1} = x = y = z$ . Projection of $O P_{1}$ on x-axis be $O M_{1}$ . Projection of $O M_{1}$ on $L_{1}$ be $O P_{2}$ and so on. Then the value of $| P_{1} M_{1} | + | P_{2} M_{2} | ........ \infty$

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$\frac{3 \sqrt{3}}{2}$

$\frac{3}{\sqrt{2}}$

$\sqrt{\frac{3}{2}}$

$\frac{\sqrt{3}}{2}$

answer is B.

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Detailed Solution

$P_{1} (1, 1, 1)$ $P_{1} M_{1} = \sqrt{2}$

$M_{1} (1, 0, 0)$

$P_{2} (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ $P_{2} M_{2} = \frac{\sqrt{2}}{3}$

$M_{2} (\frac{1}{3}, 0, 0)$ $P_{3} M_{3} = \frac{\sqrt{2}}{9}$

Hence $P_{1} M_{1} + P_{2} M_{2} ..... = \frac{\sqrt{2}}{1 - \frac{1}{3}} = \frac{3}{\sqrt{2}}$

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