To find the equation of a circle in standard form given the endpoints of its diameter, let's go through the required steps.
### Step-by-Step Solution
1. Identify the given points:
- Endpoint 1: [tex]\((8, -1)\)[/tex]
- Endpoint 2: [tex]\((-2, 3)\)[/tex]
2. Calculate the center (midpoint) of the circle:
The center of the circle is the midpoint of the diameter, which can be calculated using the midpoint formula:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting the given points:
[tex]\[
\left( \frac{8 + (-2)}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{6}{2}, \frac{2}{2} \right) = (3, 1)
\][/tex]
Therefore, the center of the circle is [tex]\((3, 1)\)[/tex].
3. Calculate the radius of the circle:
The radius is half the length of the diameter. First, we find the length of the diameter using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the given points:
[tex]\[
\text{Distance} = \sqrt{((-2) - 8)^2 + (3 - (-1))^2} = \sqrt{(-10)^2 + (4)^2} = \sqrt{100 + 16} = \sqrt{116} = 2 \sqrt{29}
\][/tex]
Hence, the length of the diameter is [tex]\(2 \sqrt{29}\)[/tex]. The radius [tex]\(r\)[/tex] is half of this length:
[tex]\[
r = \frac{2 \sqrt{29}}{2} = \sqrt{29}
\][/tex]
4. Write the equation of the circle in standard form:
The standard form of the circle equation 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.
Substituting the center [tex]\((3, 1)\)[/tex] and the radius [tex]\(\sqrt{29}\)[/tex]:
[tex]\[
(x - 3)^2 + (y - 1)^2 = (\sqrt{29})^2
\][/tex]
Simplifying further:
[tex]\[
(x - 3)^2 + (y - 1)^2 = 29
\][/tex]
Therefore, the equation of the circle in standard form is:
[tex]\[
(x - 3)^2 + (y - 1)^2 = 29
\][/tex]
