The equation `x^(2)+y^(2)-6x+8y+25=0` represents

Options

A. a point (3, -4)
B. a pair of straight lines x=3, y=-4
C. a circle of non-zero radius
D. none of these

Explanation

To determine what the equation \( x^{2} + y^{2} - 6x + 8y + 25 = 0 \) represents, we will rewrite it in a more recognizable form.

### Step-by-Step Solution:

1. **Write the given equation:**
\[ x^{2} + y^{2} - 6x + 8y + 25 = 0 \]

2. **Rearrange the equation:**
We can rearrange the equation by moving the constant term to the right side:
\[ x^{2} + y^{2} - 6x + 8y = -25 \]

3. **Complete the square for the \( x \) terms:**
To complete the square for \( x^{2} - 6x \):
- Take half of the coefficient of \( x \) (which is -6), square it:
\[ \left(-\frac{6}{2}\right)^{2} = 9 \]
- Add and subtract 9:
\[ (x^{2} - 6x + 9) - 9 \]

4. **Complete the square for the \( y \) terms:**
To complete the square for \( y^{2} + 8y \):
- Take half of the coefficient of \( y \) (which is 8), square it:
\[ \left(\frac{8}{2}\right)^{2} = 16 \]
- Add and subtract 16:
\[ (y^{2} + 8y + 16) - 16 \]

5. **Rewrite the equation with completed squares:**
Substitute the completed squares back into the equation:
\[ (x - 3)^{2} - 9 + (y + 4)^{2} - 16 = -25 \]
Simplifying this gives:
\[ (x - 3)^{2} + (y + 4)^{2} - 25 = -25 \]
\[ (x - 3)^{2} + (y + 4)^{2} = 0 \]

6. **Analyze the equation:**
The equation \( (x - 3)^{2} + (y + 4)^{2} = 0 \) indicates that both squares must equal zero for the equation to hold true. This implies:
\[ x - 3 = 0 \quad \text{and} \quad y + 4 = 0 \]
Thus, we find:
\[ x = 3 \quad \text{and} \quad y = -4 \]

7. **Conclusion:**
The equation represents a single point, which is \( (3, -4) \).

### Final Answer:
The equation \( x^{2} + y^{2} - 6x + 8y + 25 = 0 \) represents the point \( (3, -4) \).