Region Bounded by Inequalities

CAT 2024 Slot 1 · QA · Medium · Coordinate Geometry

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In the XYXY-plane, the area, in sq. units, of the region defined by the inequalities y≥x+4y \ge x + 4 and −4≤x2+y2+4(x−y)≤0-4 \le x^2 + y^2 + 4(x - y) \le 0 is

  1. A.

    4\pi

  2. B.

    2\pi

  3. C.

    \pi

  4. D.

    3\pi

Answer

B

Explanation

Rewrite −4≤x2+y2+4x−4y≤0-4 \le x^2 + y^2 + 4x - 4y \le 0 by completing squares: −4≤(x+2)2−4+(y−2)2−4≤0-4 \le (x+2)^2 - 4 + (y-2)^2 - 4 \le 0 4≤(x+2)2+(y−2)2≤84 \le (x+2)^2 + (y-2)^2 \le 8 This region represents an annulus bounded by concentric circles centered at (−2,2)(-2, 2) with radii r=2r = 2 and R=22R = 2\sqrt{2}.

The line y=x+4y = x + 4 passes directly through the center (−2,2)(-2, 2). The inequality y≥x+4y \ge x + 4 restricts the area to half of the annulus.

Area=12π(R2−r2)=12π(8−4)=2π\text{Area} = \frac{1}{2} \pi (R^2 - r^2) = \frac{1}{2} \pi (8 - 4) = 2\pi

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