Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
If the distance of any point $P(x, y)$ from the origin is defined as $d(x, y) = \max\{|x|, |y|\}$, $d(x, y) = 2$ then the area of curve represented by the locus of point $P$ is $S$ then $[S]$ is ____.
Step-by-Step Solution
Key Concept: The condition $\max\{|x|, |y|\} = c$ describes the boundary of a square with side length $2c$ centered at the origin.
Given $\max\{|x|, |y|\} = 2$, the locus consists of all points where the maximum absolute coordinate equals 2. Case I: $|x| = 2$ gives $x = \pm 2$ with $|y| \leq 2$, forming two vertical lines. Case II: $|y| = 2$ gives $y = \pm 2$ with $|x| \leq 2$, forming two horizontal lines. These four line segments form a square with vertices at $(\pm 2, \pm 2)$, side length 4, and area 16 square units.
Correct Answer: 16