Question

The area of the closed region bounded by the equation \(| x | + | y | = 2\) in the two-dimensional plane is

• A. \( 4\pi\)
• B. 4
• C. 8
• D. \( 2\pi\)

Answer 1

The Correct Option is C

Area Enclosed by |ax| + |by| = c 

Step 1: Understanding the equation

The equation: \[ |ax| + |by| = c \] in 2‑D represents a rhombus centered at the origin.

Step 2: Intercepts

From the equation:

• x‑intercepts: \( x = \pm \frac{c}{|a|} \) (when \( y = 0 \))
• y‑intercepts: \( y = \pm \frac{c}{|b|} \) (when \( x = 0 \))

These intercepts form the vertices of the rhombus.

Step 3: Special case — square

If the intercepts are equal in magnitude, the rhombus becomes a square. In the given scenario, each diagonal measures: \[ d_1 = 4, \quad d_2 = 4 \]

Step 4: Area of the square (or rhombus)

The area of a rhombus is given by: \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \] Substitute: \[ \text{Area} = \frac{1}{2} \times 4 \times 4 = 8 \]

Final Answer:

\[ \boxed{8} \]