The equation `x^(2)+y^(2)+4x+6y+13=0` represents

Options

A. a circle
B. a pair of two straight lines
C. a pair of coincident straight lines
D. a point

Answer

To determine what the equation \( x^2 + y^2 + 4x + 6y + 13 = 0 \) represents, we can follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ x^2 + y^2 + 4x + 6y + 13 = 0 \]
We can rearrange this to isolate the constant on one side:
\[ x^2 + y^2 + 4x + 6y = -13 \]

### Step 2: Complete the square for \(x\)
To complete the square for the \(x\) terms \(x^2 + 4x\), we take half of the coefficient of \(x\), square it, and add it inside the equation:
\[ x^2 + 4x = (x + 2)^2 - 4 \]
So we rewrite the equation:
\[ (x + 2)^2 - 4 + y^2 + 6y = -13 \]

### Step 3: Complete the square for \(y\)
Next, we complete the square for the \(y\) terms \(y^2 + 6y\):
\[ y^2 + 6y = (y + 3)^2 - 9 \]
Now substitute this back into the equation:
\[ (x + 2)^2 - 4 + (y + 3)^2 - 9 = -13 \]

### Step 4: Simplify the equation
Now we simplify the equation:
\[ (x + 2)^2 + (y + 3)^2 - 13 = -13 \]
This simplifies to:
\[ (x + 2)^2 + (y + 3)^2 = 0 \]

### Step 5: Analyze the equation
The equation \( (x + 2)^2 + (y + 3)^2 = 0 \) indicates that the sum of two squares is equal to zero. This can only happen if both squares are zero:
\[ (x + 2)^2 = 0 \quad \text{and} \quad (y + 3)^2 = 0 \]
This gives us:
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ y + 3 = 0 \quad \Rightarrow \quad y = -3 \]

### Conclusion
Thus, the equation represents a single point at \((-2, -3)\).

### Final Answer
The equation \( x^2 + y^2 + 4x + 6y + 13 = 0 \) represents a point.

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