Which of the following equations describes the set of all point (x, y) in the coordinate plane that are a distance of `sqrt(3)` from the point `(2, -5)` ?

Options

A. `(x-2)^(2)+(y-5)^(2)=3`
B. `(x-2)^(2)+(y+5)^(2)=3`
C. `(x+2)^(2)+(y+5)^(2)=3`
D. `(x+2)^(2)-(y-5)^(2)=3`

Explanation

To find the equation that describes the set of all points \((x, y)\) in the coordinate plane that are a distance of \(\sqrt{3}\) from the point \((2, -5)\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the center and radius**:
- The center of the circle is given as the point \((2, -5)\).
- The radius is given as \(\sqrt{3}\).

2. **Use the standard equation of a circle**:
- The standard equation of a circle with center \((a, b)\) and radius \(r\) is:
\[ (x - a)^2 + (y - b)^2 = r^2 \]

3. **Substitute the values into the equation**:
- Here, \(a = 2\), \(b = -5\), and \(r = \sqrt{3}\).
- Substitute these values into the equation:
\[ (x - 2)^2 + (y + 5)^2 = (\sqrt{3})^2 \]

4. **Calculate \(r^2\)**:
- Since \(r = \sqrt{3}\), we have:
\[ r^2 = (\sqrt{3})^2 = 3 \]

5. **Write the final equation**:
- Now, substituting \(r^2\) into the equation gives:
\[ (x - 2)^2 + (y + 5)^2 = 3 \]

### Final Answer:
The equation that describes the set of all points \((x, y)\) that are a distance of \(\sqrt{3}\) from the point \((2, -5)\) is:
\[ (x - 2)^2 + (y + 5)^2 = 3 \]