Certainly! Here's how we can determine the correct equation for the circle given its center and radius.
1. Identify the Standard Form of a Circle's Equation:
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
2. Substitute the Given Values:
Given the center is [tex]\((3.2, -2.1)\)[/tex] and the radius is [tex]\(4.3\)[/tex], substitute these into the standard form equation:
[tex]\[
(x - 3.2)^2 + (y - (-2.1))^2 = (4.3)^2
\][/tex]
3. Simplify the Equation:
Simplify the terms inside the parentheses:
[tex]\[
(x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2
\][/tex]
4. Compare with the Given Options:
Now, we compare this equation to the given options:
- Option A: [tex]\((x - 2.1)^2 - (y + 3.2)^2 = (4.3)^2\)[/tex]
[tex]\[
(x - 2.1)^2 - (y + 3.2)^2 \quad (\text{wrong structure})
\][/tex]
- Option B: [tex]\((x + 3.2)^2 + (y - 2.1)^2 = 4.3\)[/tex]
[tex]\[
... = 4.3 \quad (\text{wrong right-hand side})
\][/tex]
- Option C: [tex]\((x + 2.1)^2 + (y - 3.2)^2 = 8.6\)[/tex]
[tex]\[
... = 8.6 \quad (\text{wrong center and right-hand side})
\][/tex]
- Option D: [tex]\((x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2\)[/tex]
[tex]\[
(x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2 \quad (\text{correct structure})
\][/tex]
Therefore, the correct equation of the circle is:
Option D: [tex]\((x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2\)[/tex]
