Question

The area enclosed by |x| + |y| = 2 is:

• A. 8 square units
• B. 4 square units
• C. 16 square units
• D. \(\sqrt{8}\) square units

Hint:

The area of the region bounded by \(|x| + |y| = a\) is \(2a^2\). In this case, a=2, so the area is \(2 \times 2^2 = 8\). This is a useful shortcut.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the area of the region defined by the equation \(|x| + |y| = 2\). This equation describes a geometric shape on the Cartesian plane.
Step 2: Key Formula or Approach:
The graph of the equation \(|x| + |y| = a\) is a rhombus (which is also a square if the axes are not rotated) centered at the origin with vertices at (a, 0), (-a, 0), (0, a), and (0, -a).
The area of a rhombus is given by the formula: Area = \(\frac{1}{2} \times d_1 \times d_2\), where \(d_1\) and \(d_2\) are the lengths of the diagonals.
Step 3: Detailed Explanation:
For the given equation, \(|x| + |y| = 2\), we have a = 2.
We can find the intercepts to determine the vertices of the shape:
1. When x = 0, \(|y| = 2\), so y = 2 or y = -2. The y-intercepts are (0, 2) and (0, -2).
2. When y = 0, \(|x| = 2\), so x = 2 or x = -2. The x-intercepts are (2, 0) and (-2, 0).
The vertices of the shape are (2, 0), (-2, 0), (0, 2), and (0, -2).
This shape is a rhombus whose diagonals lie along the x and y axes.
The length of the diagonal along the x-axis, \(d_1\), is the distance between (2, 0) and (-2, 0), which is \(2 - (-2) = 4\) units.
The length of the diagonal along the y-axis, \(d_2\), is the distance between (0, 2) and (0, -2), which is \(2 - (-2) = 4\) units.
Now, we calculate the area of the rhombus:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 4 \times 4 = 8 \] Step 4: Final Answer:
The area enclosed by the given equation is 8 square units.