SATHEE: Straight Line and Pair of Straight Lines 1 Question 70

Straight Line and Pair of Straight Lines 1 Question 70

70. For a point $P$ in the plane, let $d_{1} (P)$ and $d_{2} (P)$ be the distances of the point $P$ from the lines $x - y = 0$ and $x + y = 0$ , respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_{1} (P) + d_{2} (P) \leq 4$ , is

(2014 Adv.)

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Solution:

PLAN Distance of a point $(x_{1}, y_{1})$ from $a x + b y + c = 0$ is given by

$| \frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}} |$

Let $P (x, y)$ is the point in first quadrant.

Now, $2 \leq | \frac{x - y}{\sqrt{2}} | + | \frac{x + y}{\sqrt{2}} | \leq 4$

$2 \sqrt{2} \leq | x - y | + | x + y | \leq 4 \sqrt{2}$

Case I $x \geq y$

$2 \sqrt{2} \leq (x - y) + (x + y) \leq 4 \sqrt{2} \Rightarrow x \in [\sqrt{2}, 2 \sqrt{2}]$

Case II $x < y$

$2 \sqrt{2} \leq y - x + (x + y) \leq 4 \sqrt{2}$

$y \in [\sqrt{2}, 2 \sqrt{2}]$

$\Rightarrow A = (2 \sqrt{2})^{2} - (\sqrt{2})^{2} = 6$ sq units

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