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