Question:
The area of the closed region bounded by the equation \(| x | + | y | = 2\) in the two-dimensional plane is
The area of the closed region bounded by the equation \(| x | + | y | = 2\) in the two-dimensional plane is
Updated On: Jul 30, 2025
- \( 4\pi\)
- 4
- 8
- \( 2\pi\)
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The Correct Option is C
Solution and Explanation
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:
\[ \boxed{8} \]
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