Let's analyze the given equation step-by-step:
[tex]$(x-2)^2 + (y-3)^2 = 4$[/tex]
### Step 1: Identify the Form of the Equation
The equation is in the standard form of a circle's equation, which is:
[tex]$(x-h)^2 + (y-k)^2 = r^2$[/tex]
Here:
- [tex]\((h, k)\)[/tex] is the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
### Step 2: Determine the Center of the Circle
By comparing [tex]$(x-2)^2 + (y-3)^2 = 4$[/tex] with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex]:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 3\)[/tex]
So, the center of the circle is [tex]\((2, 3)\)[/tex].
### Step 3: Determine the Radius of the Circle
The right-hand side of the equation is 4, which represents [tex]\(r^2\)[/tex]:
- [tex]\(r^2 = 4\)[/tex]
To find [tex]\(r\)[/tex], we take the square root of both sides:
- [tex]\(r = \sqrt{4}\)[/tex]
- [tex]\(r = 2\)[/tex]
Thus, the radius of the circle is [tex]\(2\)[/tex].
### Summary
- The center of the circle is [tex]\((2, 3)\)[/tex].
- The radius of the circle is [tex]\(2\)[/tex].
To recap, we’ve extracted the following details from the circle's equation [tex]\( (x-2)^2 + (y-3)^2 = 4 \)[/tex]:
- Center [tex]\((h, k) = (2, 3)\)[/tex]
- Radius [tex]\(r = 2\)[/tex]
Let's compile all this information clearly:
- Center: (2, 3)
- Radius squared: 4
- Radius: 2
These values comprehensively describe the given circle equation.
