To find the center and the radius of the circle given by the equation [tex]\((x + 2)^2 + (y - 5)^2 = 49\)[/tex], we can follow these steps:
1. Identify the standard form of the circle's equation: The standard form of a circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius.
2. Compare the given equation to the standard form:
- The given equation is [tex]\((x + 2)^2 + (y - 5)^2 = 49\)[/tex].
- Here, [tex]\((x + 2)^2\)[/tex] can be rewritten as [tex]\((x - (-2))^2\)[/tex], thus identifying [tex]\(h = -2\)[/tex].
- Similarly, [tex]\((y - 5)^2\)[/tex] is already in the required form with [tex]\(k = 5\)[/tex].
3. Determine the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
- From the comparison, we find that [tex]\(h = -2\)[/tex] and [tex]\(k = 5\)[/tex].
4. Find the radius: The right side of the equation represents [tex]\(r^2\)[/tex]:
- We have [tex]\(r^2 = 49\)[/tex].
- To find the radius [tex]\(r\)[/tex], take the square root of 49:
[tex]\[
r = \sqrt{49} = 7
\][/tex]
So, the circle's center has coordinates [tex]\((-2, 5)\)[/tex], and the radius is [tex]\(7\)[/tex] units long.
Therefore, the completed information is:
- Center coordinates: [tex]\((-2, 5)\)[/tex]
- Radius: [tex]\(7\)[/tex] units long
