To find the equation of the circle given its center and a point on the circle, we need to follow these steps:
1. Identify the center of the circle [tex]\((h, k)\)[/tex] and substitute into [tex]\((x - h)^2 + (y - k)^2\)[/tex].
2. Calculate the radius [tex]\(r\)[/tex] using the distance formula between the center and the given point.
3. Square the radius to get [tex]\(r^2\)[/tex] and complete the standard form of the circle [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
Given:
- Center of the circle [tex]\((h, k) = (5, -4)\)[/tex]
- Point on the circle [tex]\((x_1, y_1) = (-3, 2)\)[/tex]
From these steps, we get:
1. Substitute [tex]\(h = 5\)[/tex] and [tex]\(k = -4\)[/tex] into the standard form:
[tex]\[(x - 5)^2 + (y + 4)^2 = r^2\][/tex]
2. Calculate the radius [tex]\(r\)[/tex] by finding the distance between the center and the given point:
[tex]\[
r = \sqrt{((-3) - 5)^2 + (2 - (-4))^2}
= \sqrt{(-8)^2 + (6)^2}
= \sqrt{64 + 36}
= \sqrt{100}
= 10
\][/tex]
3. Substitute [tex]\(r\)[/tex] into the equation:
[tex]\[
r^2 = 10^2 = 100
\][/tex]
Therefore, the equation of the circle in standard form is:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100
\][/tex]
Finally, filling in the blanks in the given format:
[tex]\[
(x + \boxed{-5})^2 + (y + \boxed{4})^2 = \boxed{100}
\][/tex]
