The line $x + y = 1$ meets x-axis at $A$ and y-axis at $B . P$ is the mid-point of $A B . P_{1}$ is the foot of the perpendicular from $P$ to $O A$ ; $M_{1}$ is that from $P_{1}$ to $O P$ ; $P_{2}$ is that from $M_{1}$ to $O A$ and so on. If $P_{n}$ denotes the nth foot of the perpendicular on $O A$ from $M_{n - 1}, and O P_{n} = k / 2^{n}$ Then $‘ k ’ = — — —$

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Detailed Solution

$x + y = 1$ meets x-axis at $A (1, 0)$ and $y -axis at B (0, 1) .$

The ordinates of P are $(1 / 2, 1 / 2)$ and $P P_{1}$ is perpendicular $O A .$

$\Rightarrow O P_{1} = P_{1} P = 1 / 2$

Equation of the line $O P$ is $y = x .$

We have ${(O M_{n - 1})}^{2} = {(O P_{n})}^{2} + {(P_{n} M_{n - 1})}^{2}$

$= 2 {(O P_{n})}^{2} = 2 p_{n}^{2} (say)$

$\Rightarrow p_{n}^{2} = \frac{1}{4} p_{n - 1}^{2} \Rightarrow p_{n} = \frac{1}{2} p_{n - 1}$

$∴ O P_{n} = p_{n} = \frac{1}{2} p_{n - 1} = \frac{1}{2^{2}} p_{n - 2} = \dots . . \frac{1}{2^{n - 1}} p_{1} = \frac{1}{2^{n}}$

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