Find the centre and radius of the circle `(x+2)^(2)+(y+3)^(2)=5`

Options

A. `(-2,-3)` and `sqrt(5)`
B. `(2,3)` and `sqrt(5)`
C. `(-3,-2)` and `5`
D. none of these

Explanation

To find the center and radius of the circle given by the equation \((x + 2)^2 + (y + 3)^2 = 5\), we will follow these steps:

### Step 1: Identify the standard form of the circle's equation
The standard form of a circle's equation is:
\[ (x - \alpha)^2 + (y - \beta)^2 = r^2 \]
where \((\alpha, \beta)\) is the center of the circle and \(r\) is the radius.

### Step 2: Rewrite the given equation
The given equation is:
\[ (x + 2)^2 + (y + 3)^2 = 5 \]
We can rewrite it to match the standard form:
\[ (x - (-2))^2 + (y - (-3))^2 = 5 \]

### Step 3: Compare with the standard form
From the rewritten equation, we can identify:
- \(\alpha = -2\)
- \(\beta = -3\)
- \(r^2 = 5\)

### Step 4: Determine the center
The center of the circle is given by the coordinates \((\alpha, \beta)\):
\[ \text{Center} = (-2, -3) \]

### Step 5: Determine the radius
To find the radius \(r\), we take the square root of \(r^2\):
\[ r = \sqrt{5} \]

### Final Answer
- The center of the circle is \((-2, -3)\).
- The radius of the circle is \(\sqrt{5}\).

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