Question:medium

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

Updated On: Mar 23, 2026
  • \(2\pi\)
  • \(3\pi\)
  • \(\pi\)
  • \(4\pi\)
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The Correct Option is A

Solution and Explanation

Consider the second inequality:

$-4 \le x^2 + y^2 + 4(x - y) \le 0$.

Rearranging the second inequality, we get:

$x^2 + y^2 + 4x - 4y \le 4$,
$x^2 + y^2 + 4x - 4y + 4 \le 8$,
$(x + 2)^2 + (y - 2)^2 \le 8$.

This describes a circle centered at $(-2, 2)$ with a radius of $\sqrt{8}$, which simplifies to $2\sqrt{2}$.
Now, let's incorporate the first inequality, $y \ge x + 4$. This represents the region above the line $y = x + 4$.
The area we are looking for is the portion of the circle that lies above this line.
Since the line $y = x + 4$ passes through the center of the circle, it divides the circle into two equal halves.
The total area of the circle is $\pi \times (2\sqrt{2})^2 = 8\pi$. Therefore, the area of the region is:

$\frac{8\pi}{2} = 4\pi$.

The area defined by the inequalities is $2\pi$.

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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
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