Let's determine the graph represented by the equation [tex]\((x-1)^2 + (y-2)^2 = 4\)[/tex].
1. Rewrite the 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]
where [tex]\((h, k)\)[/tex] is the center of the circle, and [tex]\(r\)[/tex] is the radius.
2. Identify the Center:
By comparing the given equation with the standard form, we can see that:
[tex]\[
h = 1, \quad k = 2
\][/tex]
Therefore, the center of the circle is [tex]\((1, 2)\)[/tex].
3. Determine the Radius:
The right side of the equation is [tex]\(4\)[/tex], which represents [tex]\(r^2\)[/tex]. To find the radius [tex]\(r\)[/tex], we take the square root:
[tex]\[
r^2 = 4 \implies r = \sqrt{4} = 2
\][/tex]
4. Summary:
The equation [tex]\((x-1)^2 + (y-2)^2 = 4\)[/tex] represents a circle with:
- Center at [tex]\((1, 2)\)[/tex]
- Radius [tex]\(2\)[/tex]
Upon interpreting these values, we can conclude that the graph is a circle centered at the point [tex]\((1, 2)\)[/tex] with a radius of [tex]\(2\)[/tex].
Any graphical representation showing a circle with these characteristics would be the correct graph for the given equation.
