To derive the equation of the circle given its center and a point it passes through, 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. Calculate the radius of the circle:
We know one point on the circle is [tex]\((-3, 2)\)[/tex]. To find the radius, use the distance formula between the center [tex]\((5, -4)\)[/tex] and the point [tex]\((-3, 2)\)[/tex]:
[tex]\[
\text{radius} = \sqrt{(5 - (-3))^2 + (-4 - 2)^2} = \sqrt{(5 + 3)^2 + (-4 - 2)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\][/tex]
3. Write the standard form of the equation of the circle:
For a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex], the equation in standard form is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting [tex]\(h = 5\)[/tex], [tex]\(k = -4\)[/tex], and [tex]\(r = 10\)[/tex]:
[tex]\[
(x - 5)^2 + (y - (-4))^2 = 10^2
\][/tex]
Simplifying the equation:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100
\][/tex]
4. Convert the equation to fit the required format:
In the version requested, we need to represent the center coordinates with the opposite signs since the standard form is [tex]\((x + h)^2 + (y + k)^2 = r^2\)[/tex].
Thus:
[tex]\[
(x - 5)^2 + (y + 4)^2 = 100
\][/tex]
Hence, the values to be filled in are:
- For [tex]\(h\)[/tex]: [tex]\(-5\)[/tex]
- For [tex]\(k\)[/tex]: [tex]\(4\)[/tex]
- For [tex]\(r^2\)[/tex]: [tex]\(100\)[/tex]
So, the equation of the circle is:
[tex]\[
(x + (-5))^2 + (y + 4)^2 = 100
\][/tex]
Therefore, the equation of this circle is [tex]\((x+\boxed{-5})^2+(y+\boxed{4})^2=\boxed{100}\)[/tex].
