To determine the graph of the equation [tex]\((x-1)^2 + (y-2)^2 = 4\)[/tex], let's go through the steps to understand what this equation represents.
1. Identify the Type of Equation:
The given equation [tex]\((x-1)^2 + (y-2)^2 = 4\)[/tex] is in the standard form of a circle's equation [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
2. Determine the Center of the Circle:
By comparing [tex]\((x-1)^2 + (y-2)^2 = 4\)[/tex] with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we can see that:
- The center of the circle, [tex]\((h, k)\)[/tex], is [tex]\((1, 2)\)[/tex].
3. Determine the Radius of the Circle:
The radius [tex]\(r\)[/tex] can be determined from the term on the right side of the equation.
- The constant on the right side of the equation is 4, which represents [tex]\(r^2\)[/tex] (the square of the radius).
- To find the radius [tex]\(r\)[/tex], take the square root of 4:
[tex]\(r = \sqrt{4} = 2\)[/tex].
4. Summary of Circle's Properties:
- Center: [tex]\((1, 2)\)[/tex]
- Radius: 2
Given these properties, the graph of the equation [tex]\((x-1)^2 + (y-2)^2 = 4\)[/tex] represents a circle with a center at the point [tex]\((1, 2)\)[/tex] and a radius of 2 units. The circle will be centered at [tex]\((1, 2)\)[/tex] and points on the circle will be 2 units away from this center in all directions.
