Let's carefully analyze the equation of the circle given by [tex]\((x+3)^2 + (y-4)^2 = 64\)[/tex].
The standard form of 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.
Now, let's rewrite the given equation to compare it with the standard form:
[tex]\[
(x + 3)^2 + (y - 4)^2 = 64
\][/tex]
Notice that we can rewrite [tex]\((x + 3)\)[/tex] as [tex]\((x - (-3))\)[/tex]. So, we have:
[tex]\[
(x - (-3))^2 + (y - 4)^2 = 64
\][/tex]
From this, we can see that the center [tex]\((h, k)\)[/tex] of the circle is [tex]\((-3, 4)\)[/tex].
Next, we compare the right side of the equation:
[tex]\[
64
\][/tex]
This represents [tex]\(r^2\)[/tex]. To find the radius [tex]\(r\)[/tex], we take the square root of 64:
[tex]\[
r = \sqrt{64} = 8
\][/tex]
Therefore, the center of the circle is [tex]\((-3, 4)\)[/tex] and the radius is 8 meters.
The correct choice among the options given is:
[tex]\[ \text{Center at } (-3, 4) ; r = 8 \][/tex]
