To find the equation of a circle centered at the point [tex]\((5, -4)\)[/tex] and passing through the point [tex]\((-3, 2)\)[/tex], we follow these steps:
1. Determine the radius:
- The radius of a circle can be found using the distance formula between the center and any point on the circle.
- The distance formula is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the center of the circle [tex]\((5, -4)\)[/tex], and [tex]\((x_2, y_2)\)[/tex] is the point [tex]\((-3, 2)\)[/tex].
Plugging in these values:
[tex]\[
d = \sqrt{((-3) - 5)^2 + (2 - (-4))^2} = \sqrt{(-8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\][/tex]
So, the radius [tex]\(r\)[/tex] is 10 units.
2. Formulate the standard equation:
- The equation of a circle 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, and [tex]\(r\)[/tex] is the radius.
Substituting [tex]\(h = 5\)[/tex], [tex]\(k = -4\)[/tex], and [tex]\(r = 10\)[/tex], the equation becomes:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 10^2
\][/tex]
3. Simplify [tex]\(r^2\)[/tex]:
- [tex]\(r^2 = 10^2 = 100\)[/tex]
Putting it all together, we get:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100
\][/tex]
To match the format provided in the problem:
The equation of this circle is [tex]\((x + \boxed{-5})^2 + (y + \boxed{4})^2 = \boxed{100}\)[/tex].
