To determine the equation of the circle, we need to follow these steps:
1. Identify the center of the circle: The center of the circle is given as [tex]\((5, -4)\)[/tex].
2. Identify a point on the circle: The circle passes through the point [tex]\((-3, 2)\)[/tex].
3. Calculate the radius of the circle:
The radius [tex]\(r\)[/tex] can be found using the distance formula between the center [tex]\((5, -4)\)[/tex] and the point on the circle [tex]\((-3, 2)\)[/tex]. However, in this case, it’s already provided that the radius squared ([tex]\(r^2\)[/tex]) is [tex]\(100.0\)[/tex].
4. Equation of the circle:
The general 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]
For the given circle, the center [tex]\((h, k)\)[/tex] is [tex]\((5, -4)\)[/tex], hence:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100
\][/tex]
To make the given equation fit the form [tex]\((x + \square)^2 + (y + \square)^2 = \square \)[/tex]:
[tex]\(\boxed{-5}\)[/tex] for [tex]\(h\)[/tex] (because [tex]\((x + (-5))\)[/tex] simplifies to [tex]\((x - 5)\)[/tex]),
[tex]\(\boxed{4}\)[/tex] for [tex]\(k\)[/tex] (because [tex]\((y + (4))\)[/tex] simplifies to [tex]\((y + 4)\)[/tex]),
[tex]\(\boxed{100.0}\)[/tex] for [tex]\(r^2\)[/tex].
So, the equation of the circle is:
[tex]\[
(x + (-5))^2 + (y + 4)^2 = 100.0
\][/tex]
If we transcribe this equation into the designated format, we get:
[tex]\[
(x + (-5))^2 + (y + 4)^2 = 100.0
\][/tex]
This can be rewritten as:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100.0
\][/tex]
Therefore, the answer is:
[tex]\[
(x + \boxed{-5})^2 + (y + \boxed{4})^2 = \boxed{100.0}
\][/tex]
