Sure, let's write out the detailed solution step-by-step.
1. Identify the center of the circle:
The center of the circle is given by the coordinates [tex]\((-3,-4)\)[/tex].
Therefore, [tex]\( h = -3 \)[/tex] and [tex]\( k = -4 \)[/tex].
2. Determine the radius of the circle:
The endpoints of the diameter of the circle are given as [tex]\((0,2)\)[/tex] and [tex]\((-6,-10)\)[/tex].
To find the radius, we first find the length of the diameter using the distance formula:
[tex]\[
\text{Diameter length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the given points [tex]\((x_1, y_1) = (0, 2)\)[/tex] and [tex]\((x_2, y_2) = (-6, -10)\)[/tex]:
[tex]\[
\text{Diameter length} = \sqrt{((-6 - 0)^2 + (-10 - 2)^2)} = \sqrt{((-6)^2 + (-12)^2)} = \sqrt{36 + 144} = \sqrt{180}
\][/tex]
Hence, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{\sqrt{180}}{2} = \sqrt{45}
\][/tex]
[tex]\[
r^2 = (\sqrt{45})^2 = 45
\][/tex]
3. Formulate the equation of the circle:
The standard form of the equation of a circle with center [tex]\((h,k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting the values [tex]\( h = -3 \)[/tex], [tex]\( k = -4 \)[/tex], and [tex]\( r^2 = 45\)[/tex]:
[tex]\[
(x - (-3))^2 + (y - (-4))^2 = 45
\][/tex]
Simplifying the double negatives:
[tex]\[
(x + 3)^2 + (y + 4)^2 = 45
\][/tex]
4. Fill in the blanks with the appropriate values:
[tex]\[
(x - (\boxed{-3}))^2 + (y - (\boxed{-4}))^2 = \boxed{45}
\][/tex]
Hence, the equation of the circle is:
[tex]\[
(x + 3)^2 + (y + 4)^2 = 45
\][/tex]
