Understanding the Curve |x| + |y| = 1

The question asks for the area enclosed by the curve defined by the equation $|x| + |y| = 1$. This equation involves absolute values, which means we need to consider different cases depending on the signs of $x$ and $y$.

Analyzing the Equation in Different Quadrants

We can break down the equation $|x| + |y| = 1$ into four parts, one for each quadrant:

• First Quadrant ($x \ge 0, y \ge 0$): Here, $|x| = x$ and $|y| = y$. The equation becomes $x + y = 1$. This is a line segment connecting the points (1, 0) on the x-axis and (0, 1) on the y-axis.
• Second Quadrant ($x < 0, y \ge 0$): Here, $|x| = -x$ and $|y| = y$. The equation becomes $-x + y = 1$. This is a line segment connecting the points (0, 1) on the y-axis and (-1, 0) on the x-axis.
• Third Quadrant ($x < 0, y < 0$): Here, $|x| = -x$ and $|y| = -y$. The equation becomes $-x - y = 1$, or $x + y = -1$. This is a line segment connecting the points (-1, 0) on the x-axis and (0, -1) on the y-axis.
• Fourth Quadrant ($x \ge 0, y < 0$): Here, $|x| = x$ and $|y| = -y$. The equation becomes $x - y = 1$. This is a line segment connecting the points (0, -1) on the y-axis and (1, 0) on the x-axis.

Identifying the Geometric Shape

Plotting these line segments, we can see they form a closed shape. The vertices of this shape are at (1, 0), (0, 1), (-1, 0), and (0, -1). This geometric figure is a square, rotated by 45 degrees, centered at the origin.

Calculating the Area of the Square

We can calculate the area of this square using its diagonals. The diagonals connect opposite vertices:

• Diagonal 1 connects (-1, 0) and (1, 0). Its length is $1 - (-1) = 2$.
• Diagonal 2 connects (0, -1) and (0, 1). Its length is $1 - (-1) = 2$.

The area of a square can be calculated using the formula: Area = $ \frac{1}{2} \times (\text{length of diagonal}_1) \times (\text{length of diagonal}_2) $

Substituting the lengths of the diagonals: Area = $ \frac{1}{2} \times 2 \times 2 $

Area = $ \frac{1}{2} \times 4 = 2 $

Alternatively, we can find the side length of the square. Let's calculate the distance between two adjacent vertices, for example, (1, 0) and (0, 1), using the distance formula $ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $: Side length $ s = \sqrt{(0-1)^2 + (1-0)^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} $. The area of the square is $ s^2 $: Area = $ (\sqrt{2})^2 = 2 $.

Conclusion

Both methods confirm that the area enclosed by the curve $|x| + |y| = 1$ is 2 square units. This corresponds to option 4.