The graph of the equation `x^2+y^2=0` in the three dimensional space is (A) x-axis (B) y-axis (C) z-axis (D) xy-plane

Options

A. x-axis
B. y-axis
C. z-axis
D. XY-plane

Answer

To solve the problem of determining the graph of the equation \(x^2 + y^2 = 0\) in three-dimensional space, we can follow these steps:

### Step 1: Analyze the Equation
The equation given is \(x^2 + y^2 = 0\). This equation implies that both \(x^2\) and \(y^2\) must be equal to zero because the sum of two non-negative numbers can only be zero if both numbers are zero.

### Step 2: Solve for \(x\) and \(y\)
From \(x^2 = 0\), we find:
\[ x = 0 \]
From \(y^2 = 0\), we find:
\[ y = 0 \]

### Step 3: Consider the \(z\) Coordinate
The equation does not involve \(z\), which means that \(z\) can take any value. Therefore, the points that satisfy the equation \(x^2 + y^2 = 0\) are of the form:
\[ (0, 0, z) \]
where \(z\) can be any real number.

### Step 4: Interpret the Result
The set of points \((0, 0, z)\) for all values of \(z\) describes a line along the \(z\)-axis in three-dimensional space.

### Conclusion
Thus, the graph of the equation \(x^2 + y^2 = 0\) in three-dimensional space is the \(z\)-axis.

### Final Answer
The correct option is (C) z-axis.

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