To determine the correct equation for the circle [tex]\( C \)[/tex] with endpoints [tex]\( J(-8, 9) \)[/tex] and [tex]\( K(-2, -5) \)[/tex] of its diameter, we need to follow these steps:
1. Find the center of the circle, which is the midpoint of the diameter.
The midpoint [tex]\((x, y)\)[/tex] of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated as:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For endpoints [tex]\( J(-8, 9) \)[/tex] and [tex]\( K(-2, -5) \)[/tex], we have:
[tex]\[
\text{Center } = \left( \frac{-8 + (-2)}{2}, \frac{9 + (-5)}{2} \right) = \left( \frac{-10}{2}, \frac{4}{2} \right) = (-5, 2)
\][/tex]
2. Calculate the radius of the circle.
The radius is half the length of the diameter. The length of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For [tex]\( J(-8, 9) \)[/tex] and [tex]\( K(-2, -5) \)[/tex], we find:
[tex]\[
\text{Distance} = \sqrt{((-2) - (-8))^2 + ((-5) - 9)^2} = \sqrt{(6)^2 + (-14)^2} = \sqrt{36 + 196} = \sqrt{232}
\][/tex]
Therefore, the radius [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{\sqrt{232}}{2} = \sqrt{58}
\][/tex]
3. Write the equation of the circle.
The general 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]
Here, the center is [tex]\((-5, 2)\)[/tex] and the radius [tex]\( r = \sqrt{58} \)[/tex]. Then:
[tex]\[
(x + 5)^2 + (y - 2)^2 = (\sqrt{58})^2 = 58
\][/tex]
So, the equation of the circle [tex]\( C \)[/tex] is:
[tex]\[
(x + 5)^2 + (y - 2)^2 = 58
\][/tex]
Therefore, the correct choice is:
C. [tex]\((x+5)^2+(y-2)^2=58\)[/tex]
