To determine the center and radius of the circle given the equation [tex]\( (x+4)^2+(y-5)^2=121 \)[/tex], we follow these steps:
1. Identify the standard form of the circle’s equation:
The general form for the equation of a circle is [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. Compare the given equation to the standard form:
The given equation is [tex]\( (x + 4)^2 + (y - 5)^2 = 121 \)[/tex]. To match this with [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex], note the following:
- The term [tex]\( (x + 4)^2 \)[/tex] can be rewritten as [tex]\( (x - (-4))^2 \)[/tex]. This tells us [tex]\( h = -4 \)[/tex].
- The term [tex]\( (y - 5)^2 \)[/tex] is already in the form required, so [tex]\( k = 5 \)[/tex].
- The right side of the equation equals 121, which is in the form [tex]\( r^2 = 121 \)[/tex].
3. Determine the center:
The center of the circle is given by [tex]\( (h, k) \)[/tex]. Thus, from the values identified:
[tex]\[
center = (-4, 5)
\][/tex]
4. Determine the radius:
The right side of the equation is [tex]\( r^2 = 121 \)[/tex]. Thus, to find [tex]\( r \)[/tex], we take the square root of 121:
[tex]\[
r = \sqrt{121} = 11
\][/tex]
Therefore, the center of the circle is [tex]\((-4, 5)\)[/tex] and the radius is 11.
Hence, the correct choice from the given answers is:
- center: [tex]\((-4, 5)\)[/tex]; radius: 11
