Equation of the circle is
.
Solution:
The endpoints of the diameter of a circle are (–8, –6) and (–4, –14).
Center of the circle = Mid point of the diameter
Mid point formula:
![$P(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)](https://tex.z-dn.net/?f=%24P%28x%2C%20y%29%3D%5Cleft%28%5Cfrac%7Bx_%7B1%7D%2Bx_%7B2%7D%7D%7B2%7D%2C%20%5Cfrac%7By_%7B1%7D%2By_%7B2%7D%7D%7B2%7D%5Cright%29)
Here, ![x_1=-8, y_1=-6, x_2=-4, y_2=-14](https://tex.z-dn.net/?f=x_1%3D-8%2C%20y_1%3D-6%2C%20x_2%3D-4%2C%20y_2%3D-14)
![$P(x, y) =\left(\frac{-8-4}{2}, \frac{-6-14}{2}\right)](https://tex.z-dn.net/?f=%24P%28x%2C%20y%29%20%3D%5Cleft%28%5Cfrac%7B-8-4%7D%7B2%7D%2C%20%5Cfrac%7B-6-14%7D%7B2%7D%5Cright%29)
![$P(x, y) =\left(\frac{-12}{2}, \frac{-20}{2}\right)](https://tex.z-dn.net/?f=%24P%28x%2C%20y%29%20%3D%5Cleft%28%5Cfrac%7B-12%7D%7B2%7D%2C%20%5Cfrac%7B-20%7D%7B2%7D%5Cright%29)
![$P(x, y) =(-6, -10)](https://tex.z-dn.net/?f=%24P%28x%2C%20y%29%20%3D%28-6%2C%20-10%29)
Center of the circle = (–6, –10)
Radius is the distance between center and any endpoint of the diameter.
To calculate the radius using distance formula.
![r=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%5Cleft%28x_%7B2%7D-x_%7B1%7D%5Cright%29%5E%7B2%7D%2B%5Cleft%28y_%7B2%7D-y_%7B1%7D%5Cright%29%5E%7B2%7D%7D)
Here, ![x_1=-6, y_1=-10, x_2=-8, y_2=-6](https://tex.z-dn.net/?f=x_1%3D-6%2C%20y_1%3D-10%2C%20x_2%3D-8%2C%20y_2%3D-6)
![r=\sqrt{\left(-8-(-6)\right)^{2}+\left(-6-(-10)}\right)^{2}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%5Cleft%28-8-%28-6%29%5Cright%29%5E%7B2%7D%2B%5Cleft%28-6-%28-10%29%7D%5Cright%29%5E%7B2%7D%7D)
![r=\sqrt{(-8+6)^{2}+(-6+10)^{2}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%28-8%2B6%29%5E%7B2%7D%2B%28-6%2B10%29%5E%7B2%7D%7D)
![r=\sqrt{(-2)^{2}+(4)^{2}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%28-2%29%5E%7B2%7D%2B%284%29%5E%7B2%7D%7D)
units
The standard form of the equation of a circle is
, where (a, b) are center and r is the radius.
Here, center = (–6, –10) and
![(x-(-6))^{2}+(y-(-10))^{2}={(\sqrt{20})} ^{2}](https://tex.z-dn.net/?f=%28x-%28-6%29%29%5E%7B2%7D%2B%28y-%28-10%29%29%5E%7B2%7D%3D%7B%28%5Csqrt%7B20%7D%29%7D%20%5E%7B2%7D)
![(x+6)^{2}+(y+10)^{2}=20](https://tex.z-dn.net/?f=%28x%2B6%29%5E%7B2%7D%2B%28y%2B10%29%5E%7B2%7D%3D20)
Equation of the circle is
.