IF sets A and B are defined as `A={(x,y):y-(1)/(x),0ne x in R}`
`B= {(x,y):y=-x,x in R},`then

Options

A. `A cap B =A `
B. `A cap B =B `
C. `A cap B =phi `
D. none of these

Answer

To solve the problem, we need to find the intersection of the two sets A and B defined as follows:

- Set A: \( A = \{(x, y) : y = \frac{1}{x}, x \neq 0, x \in \mathbb{R}\} \)
- Set B: \( B = \{(x, y) : y = -x, x \in \mathbb{R}\} \)

### Step 1: Set up the equations
From the definitions of sets A and B, we have:
1. From set A: \( y = \frac{1}{x} \)
2. From set B: \( y = -x \)

### Step 2: Equate the two equations
To find the intersection \( A \cap B \), we need to set the two equations equal to each other:
\[ \frac{1}{x} = -x \]

### Step 3: Clear the fraction
Multiply both sides by \( x \) (noting that \( x \neq 0 \)):
\[ 1 = -x^2 \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ x^2 = -1 \]

### Step 5: Analyze the equation
The equation \( x^2 = -1 \) has no real solutions since the square of a real number cannot be negative.

### Conclusion
Since there are no real values of \( x \) that satisfy both equations, the intersection of sets A and B is empty. Therefore:
\[ A \cap B = \emptyset \]

### Final Answer
The intersection of sets A and B is \( \emptyset \) (the empty set).

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