To determine the center and radius of the circle given by the equation [tex]\((x-5)^2 + (y-1)^2 = 30\)[/tex], let's analyze it step-by-step:
1. Identify the standard form of the circle's equation:
A circle's equation in standard form is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where:
- [tex]\((h, k)\)[/tex] is the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
2. Compare the given equation with the standard form:
The given equation is [tex]\((x-5)^2 + (y-1)^2 = 30\)[/tex].
- By comparing, we see that [tex]\(h = 5\)[/tex] and [tex]\(k = 1\)[/tex].
- Hence, the center of the circle is [tex]\((5, 1)\)[/tex].
3. Determine the radius:
- According to the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], the term on the right-hand side is [tex]\(r^2\)[/tex].
- Here, [tex]\(r^2 = 30\)[/tex].
- To find [tex]\(r\)[/tex], we take the square root of 30: [tex]\(r = \sqrt{30}\)[/tex].
- The approximate value of [tex]\(\sqrt{30}\)[/tex] is [tex]\(5.477225575051661\)[/tex] (not asked for but useful for approximate understanding).
4. Conclusion:
- The correct center of the circle is [tex]\((5, 1)\)[/tex].
- The correct radius of the circle is [tex]\(\sqrt{30}\)[/tex].
Therefore, the correct answer is:
Center: [tex]\((5, 1)\)[/tex]. Radius: [tex]\(\sqrt{30}\)[/tex].
