To find the equation of the circle when given the endpoints of a diameter, you can follow these steps:
1. Find the center of the circle: Since the center is the midpoint of the diameter, you can calculate it using the midpoint formula:
In general, the midpoint [tex]\( M \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
For the given points [tex]\( (2, 10) \)[/tex] and [tex]\( (8, -14) \)[/tex]:
[tex]\[ M = \left(\frac{2 + 8}{2}, \frac{10 + (-14)}{2}\right) \][/tex]
[tex]\[ M = (5, -2) \][/tex]
So, the center of the circle [tex]\( C \)[/tex] is at [tex]\( (5, -2) \)[/tex].
2. Calculate the radius of the circle: The radius is half of the distance between the endpoints of the diameter. The distance between two points is given by the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For the given points:
[tex]\[
d = \sqrt{(8 - 2)^2 + (-14 - 10)^2}
\][/tex]
[tex]\[
d = \sqrt{6^2 + (-24)^2}
\][/tex]
[tex]\[
d = \sqrt{36 + 576}
\][/tex]
[tex]\[
d = \sqrt{612}
\][/tex]
Since this distance is the diameter, to find the radius [tex]\( r \)[/tex], we divide by two:
[tex]\[
r = \frac{d}{2} = \frac{\sqrt{612}}{2} = \sqrt{153}
\][/tex]
3. Write the equation of the circle: The standard form of the equation of a circle with center at [tex]\( (h, k) \)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\( h = 5 \)[/tex], [tex]\( k = -2 \)[/tex], and [tex]\( r = \sqrt{153} \)[/tex]. Therefore, substituting these values in, we get:
[tex]\[
(x - 5)^2 + (y + 2)^2 = (\sqrt{153})^2
\][/tex]
[tex]\[
(x - 5)^2 + (y + 2)^2 = 153
\][/tex]
This is the equation of the circle with the diameter endpoints [tex]\( (2, 10) \)[/tex] and [tex]\( (8, -14) \)[/tex].
