For a point P in the plane, let d1(P) and d2(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 ≤ d1(P) + d2(P) ≤ 4, is :

Engineering

Mathematics

Length of Perpendicular

Question

For a point P in the plane, let d₁(P) and d₂(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 ≤ d₁(P) + d₂(P) ≤ 4, is :

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$∴ 2 \leq | \frac{h - k}{\sqrt{2}} | + | \frac{h + k}{\sqrt{2}} | \leq 4$

$2 \sqrt{2} \leq | h - k | + | h + k | \leq 4 \sqrt{2}$

Required area = $\frac{1}{4} ({(4 \sqrt{2})}^{2} - {(2 \sqrt{2})}^{2}) = 6$ (sq. units)