If `|x + y| = |x - y|` then the number of ordered pairs of `(x , y)` which satisfy the given condition is :

Options

A. a. 1
B. b. 4
C. c. infinite
D. d. none of these

Answer

To solve the equation \( |x + y| = |x - y| \), we can analyze it step by step.

### Step 1: Understanding the Absolute Value Equation
The equation \( |x + y| = |x - y| \) means that the expressions inside the absolute values can either be equal or opposite. Therefore, we can break this down into two cases:

1. \( x + y = x - y \)
2. \( x + y = -(x - y) \)

### Step 2: Solving the First Case
For the first case:
\[ x + y = x - y \]
Subtract \( x \) from both sides:
\[ y = -y \]
Adding \( y \) to both sides gives:
\[ 2y = 0 \implies y = 0 \]

### Step 3: Solving the Second Case
For the second case:
\[ x + y = -(x - y) \]
This simplifies to:
\[ x + y = -x + y \]
Adding \( x \) to both sides results in:
\[ 2x + y = 0 \implies y = -2x \]

### Step 4: Finding Ordered Pairs
From the first case, we found that \( y = 0 \). This means \( x \) can be any real number. Thus, the ordered pairs are of the form \( (x, 0) \) where \( x \) can take any value.

From the second case, we have \( y = -2x \). Here, for every real number \( x \), there is a corresponding \( y \). Thus, the ordered pairs are of the form \( (x, -2x) \).

### Step 5: Conclusion
Since both cases yield an infinite number of ordered pairs, we conclude that the total number of ordered pairs \( (x, y) \) that satisfy the given condition is infinite.

Thus, the answer is:
\[ \text{The number of ordered pairs } (x, y) \text{ is infinite.} \]