To determine the equation of a circle with a given center and a point on the circle, we need to identify three parameters in the standard form of the circle's equation:
[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.
Step-by-step solution:
1. Identify the center of the circle:
The center of the circle is given as [tex]\((h, k) = (5, -4)\)[/tex].
2. Identify a point on the circle:
The point on the circle is [tex]\((x_1, y_1) = (-3, 2)\)[/tex].
3. Calculate the radius:
The radius [tex]\(r\)[/tex] can be found using the distance formula between the center [tex]\((5, -4)\)[/tex] and the given point [tex]\((-3, 2)\)[/tex]:
[tex]\[
r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\][/tex]
Plugging in the values:
[tex]\[
r = \sqrt{((-3 - 5)^2 + (2 + 4)^2)} = \sqrt{((-8)^2 + (6)^2)} = \sqrt{64 + 36} = \sqrt{100} = 10
\][/tex]
4. Form the equation of the circle:
Using the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], and substituting [tex]\(h = 5\)[/tex], [tex]\(k = -4\)[/tex], and [tex]\(r = 10\)[/tex], the equation becomes:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 10^2 = 100
\][/tex]
5. Fit the equation into the required format:
The standard form given in the question is [tex]\((x + \square)^2 + (y + \square)^2 = \square\)[/tex].
To match this, consider:
[tex]\[
(x - 5)^2 = (x + (-5))^2
\][/tex]
and
[tex]\[
(y + 4)^2 = (y + (-4))^2.
\][/tex]
6. Final values for the equation:
Substitute the appropriate values into the given format:
[tex]\[
(x + (-5))^2 + (y + (-4))^2 = 100
\][/tex]
Thus, the correct values to fill in the blanks are:
[tex]\((x + \boxed{-5})^2 + (y + \boxed{-4})^2 = \boxed{100}\)[/tex]
