If the distance of any point (x, y) from origin is defined as `d(x,y)="m ax"{|x|,|y|}`, then the locus of the point `(x,y)`, where `(x,y)=1` is:

Options

A. A circle
B. A square
C. A triangle
D. None of these

Answer

To find the locus of the point \((x, y)\) where the distance from the origin is defined as \(d(x, y) = \max\{|x|, |y|\} = 1\), we will follow these steps:

### Step 1: Set up the equation
Given the definition of distance, we have:
\[ \max\{|x|, |y|\} = 1 \]

### Step 2: Analyze the maximum function
The equation \(\max\{|x|, |y|\} = 1\) implies two conditions:
1. \(|x| \leq 1\)
2. \(|y| \leq 1\)

### Step 3: Break down the conditions
From the above conditions, we can derive:
- For \(|x| \leq 1\), we have:
\[ -1 \leq x \leq 1 \]
- For \(|y| \leq 1\), we have:
\[ -1 \leq y \leq 1 \]

### Step 4: Identify the boundaries
The conditions \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\) define a square in the coordinate plane. The vertices of this square are:
- \((-1, -1)\)
- \((1, -1)\)
- \((1, 1)\)
- \((-1, 1)\)

### Step 5: Draw the locus
When we plot these points on the Cartesian plane, we see that they form a square with sides parallel to the axes, centered at the origin.

### Conclusion
The locus of the point \((x, y)\) where \(d(x, y) = 1\) is a square with vertices at \((-1, -1)\), \((1, -1)\), \((1, 1)\), and \((-1, 1)\).

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