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Properties of Set Operation

Let a P , OA ...

Question

Let $a (P, O A) \leq m i n d (P, A B), d (P, B C), d (P, O C)$ where d denotes the distance from the point to the corresponding line and S be the region consisting of all those points P inside the rectangle OABC such that $O = (0, 0), A = (3, 0), B = (3, 2) a n d C = (0, 2),$ which satisfy the above relation, then area of the region S is

2 sq unit

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3 sq unit

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4 sq unit

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1 sq unit

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Solution

The correct option is A 2 sq unit
Let (x,y) be the coordinates of P'

Case 1: $0 \leq x \leq 1 a n d 0 \leq y \leq 1$
$m i n . d (P, A B), d (P, B C), d (P, O C) = d (P, O C)$
$∴ d (P, O A) \leq d (P, O C)$
$y \leq x$
Case 2: $1 \leq x \leq 2$
$m i n . d (P, A B), d (P, B C), d (P, O C) = d (P, B C)$
$∴ d (P, O A) \leq d (P, B C)$
$\Rightarrow y \leq 3 - y o r y \leq 1$
Case 3: $2 \leq x \leq 3$ and $0 \leq y \leq 1$
$m i n . d (P, A B), d (P, B C), d (P, O C) = d (P, A B)$
$∴ d (P, O A) \leq d (P, A B)$
$\Rightarrow y \leq 3 - x o r y \leq 3$
Area of trapezium OAED ub required region
$∴ A r e a = \frac{1}{2} (3 + 1) \times 1 = 2 s q u n i t$ .

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