To find the equation of a circle given the center and the endpoints of a diameter, follow these steps:
1. Identify the center of the circle:
The center of the circle is [tex]\((-3, -4)\)[/tex].
2. Calculate the radius of the circle:
Use the distance formula to find the length of the diameter first. The formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
The given endpoints of the diameter are [tex]\((0, 2)\)[/tex] and [tex]\((-6, -10)\)[/tex]. Plugging in these values:
[tex]\[
d = \sqrt{(0 - (-6))^2 + (2 - (-10))^2} = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180}
\][/tex]
The diameter is [tex]\(\sqrt{180}\)[/tex], and the radius [tex]\(r\)[/tex] is half of the diameter:
[tex]\[
r = \frac{\sqrt{180}}{2} = \sqrt{45} \approx 6.708203932499369
\][/tex]
3. Formulate the equation of the circle:
The general form of the equation of a circle is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
Here, the center [tex]\((h, k) = (-3, -4)\)[/tex], and [tex]\(r^2 = (\sqrt{45})^2 = 45\)[/tex].
Thus, the equation of the circle is:
[tex]\[
(x - (-3))^2 + (y - (-4))^2 = 45
\][/tex]
Simplifying:
[tex]\[
(x + 3)^2 + (y + 4)^2 = 45
\][/tex]
So filling in the provided boxes:
[tex]\[
\boxed{ -3 } \;\;\boxed{ -4 }\;\;\boxed{ 45 }
\][/tex]
The final equation is:
[tex]\[
(x + 3)^2 + (y + 4)^2 = 45
\][/tex]
