Respuesta :
Equation of the circle is [tex](x+6)^{2}+(y+10)^{2}=20[/tex].
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:
[tex]$P(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)[/tex]
Here, [tex]x_1=-8, y_1=-6, x_2=-4, y_2=-14[/tex]
[tex]$P(x, y) =\left(\frac{-8-4}{2}, \frac{-6-14}{2}\right)[/tex]
[tex]$P(x, y) =\left(\frac{-12}{2}, \frac{-20}{2}\right)[/tex]
[tex]$P(x, y) =(-6, -10)[/tex]
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.
[tex]r=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}[/tex]
Here, [tex]x_1=-6, y_1=-10, x_2=-8, y_2=-6[/tex]
[tex]r=\sqrt{\left(-8-(-6)\right)^{2}+\left(-6-(-10)}\right)^{2}}[/tex]
[tex]r=\sqrt{(-8+6)^{2}+(-6+10)^{2}}[/tex]
[tex]r=\sqrt{(-2)^{2}+(4)^{2}}[/tex]
[tex]r=\sqrt{20}[/tex] units
The standard form of the equation of a circle is
[tex](x-a)^{2}+(y-b)^{2}=r^{2}[/tex], where (a, b) are center and r is the radius.
Here, center = (–6, –10) and [tex]r=\sqrt{20}[/tex]
[tex](x-(-6))^{2}+(y-(-10))^{2}={(\sqrt{20})} ^{2}[/tex]
[tex](x+6)^{2}+(y+10)^{2}=20[/tex]
Equation of the circle is [tex](x+6)^{2}+(y+10)^{2}=20[/tex].
