CAT 2025 Slot 3 QA Question 27: In the $XY$-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x + 4$ and $-4 \leq x^2 + y^2 + 4(x-y) \leq 0$ is

Question

In the $XY$-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x + 4$ and $-4 \leq x^2 + y^2 + 4(x-y) \leq 0$ is

Accepted Answer

Rewrite the second inequality by completing squares.
$x^2 + y^2 + 4(x-y) = x^2 + y^2 + 4x - 4y$
$= (x+2)^2 + (y-2)^2 - 8$
$-4 \leq (x+2)^2 + (y-2)^2 - 8 \leq 0$
$\Rightarrow 4 \leq (x+2)^2 + (y-2)^2 \leq 8$
This is an annulus centered at $(-2, 2)$, with inner radius $2$ and outer radius $2\sqrt{2}$.

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

In the $XY$ -plane, the area, in sq. units, of the region defined by the inequalities $y \geq x + 4$ and $-4 \leq x^2 + y^2 + 4(x-y) \leq 0$ is